Загрузил Вадим Орлов

romero2004

реклама
J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
Low-flow limit in slot coating of dilute solutions of high
molecular weight polymer
O.J. Romero a , W.J. Suszynski b , L.E. Scriven b , M.S. Carvalho a,∗
a
Department of Mechanical Engineering, Pontificia Universidade Catolica do Rio de Janeiro, Rua Marques de Sao Vicente 225,
Gavea, Rio de Janeiro, RJ 22453-900, Brazil
b Department of Chemical Engineering & Materials Science, University of Minnesota, MN, USA
Accepted 14 March 2004
Abstract
Slot coating is a common method in the manufacture of a wide variety of products. It belongs to a class of coating method known as
premetered coating: the thickness of the coated liquid layer in principle is set by the flow rate fed to the die and the speed of the substrate
moving past, and is independent of other process variables. Thus, premetered methods are ideal for high precision coating. An important
operating limit of slot coating is the minimum thickness that can be coated at a given substrate speed, generally referred to as the low-flow
limit. The mechanism that defines this limit balances the viscous, capillary and inertial forces in the flow. Although most of the liquids coated
industrially are polymeric solutions and dispersions that are not Newtonian, previous analyses of the low-flow limit in slot coating dealt only
with Newtonian liquids. In this paper, the low-flow limit in slot coating of an extensional thickening polymer solution is examined both by
theory and by experiment. The continuity and momentum equations coupled with an algebraic non-Newtonian constitutive equation that
relates stress to the rate-of-strain and relative-rate-of-rotation tensors were solved by the Galerkin/finite element method to model the flows.
The flows themselves were visualized by video microscopy and the low-flow limit was found by observing, at given substrate speed, the
feed rate at which the flow becomes unstable and breaks up. Various solutions of low molecular weight polyethylene glycol (PEG) and high
molecular weight polyethylene oxide (PEO) in water were used in order to evaluate the effect of mildly viscoelastic behavior on the process.
At the concentration level of the high molecular weight polymer explored here, the viscoelastic behavior of the solutions could not be accessed
by oscillatory tests; the only measurable response to the addition of PEO was the rise of the apparent extensional viscosity.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Free surface flow; Slot coating; Low-flow limit; Dilute polymer solutions
1. Introduction
Slot coating is commonly used in the manufacturing of adhesive and magnetic tapes, specialty papers, imaging films,
and many other products. In this process, the coating liquid
is pumped to a coating die in which an elongated chamber
distributes it across the width of a narrow slot through which
the flow rate per unit width at the slot exit is made uniform.
Exiting the slot, the liquid fills (wholly or partially) the gap
between the adjacent die lips and the substrate translating
rapidly past them. The liquid in the gap, bounded upstream,
and downstream by gas–liquid interfaces, or menisci, forms
the coating bead, as shown in Fig. 1. The competition among
∗
Corresponding author. Fax: +55-21-3114-1165.
E-mail address: msc@mec.puc-rio.br (M.S. Carvalho).
0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jnnfm.2004.03.004
viscous, capillary and pressure forces, and in some cases inertial and elastic forces, sets the range of operating parameters in which the viscous free surface flow of the liquid can
be two-dimensional and steady, which is the desired state. In
order to sustain the coating bead at higher substrate speeds,
the gas pressure upstream of the upstream meniscus is made
lower than ambient, i.e. a slight vacuum is applied to the
upstream meniscus [2]. Slot coating belongs to a class of
coating methods known as pre-metered coating: the thickness of the coated liquid layer is set by the flow rate fed to
the coating die and the speed of the moving substrate, and
is independent of other process variables. Thus, pre-metered
methods are ideal for high precision coating. However, the
nature of the flow in the coating bead, and therefore the uniformity of the liquid layer it delivers, can be affected by
the substrate speed, the viscosity and any non-Newtonian
properties of the liquid, and the configuration of the die lips
138
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
FEED SLOT
SLOT DIE
DISTRIBUTION
CHAMBER
SUCTION
VACUUM BOX
Static contac line
Static contac line
UPSTREAM LIP
UPSTREAM
MENISCUS
DOWNSTREAM LIP
COATING
GAP
Pvac
H0
COATING BEAD
DOWNSTREAM
MENISCUS
Patm
FILM THICKNESS,
t
SUBSTRATE VELOCITY, Vw
DYNAMIC
CONTACT LINE
Fig. 1. Sketch of slot coating bead.
immediately upstream and downstream of the slot exit. The
region in the space of operating parameters of a coating process where the delivered liquid layer is adequately uniform
is usually referred to as a coating window. Knowledge of
coating window’s for different coating methods is needed in
order to predict whether a particular method can be used to
coat a given substrate at a prescribed production rate.
Ruschak [24] analyzed the coating window of a slot
coating bead dominated by surface tension force (capillary
pressure) in the upstream and downstream menisci; Silliman [28] and Higgins and Scriven [15] took the viscous
drag of the substrate and die lips into account. More refined theoretical analyses and experiments on the limits of
operability and flow stability within those limits were made
subsequently [4,12,25]; the results are typified in Fig. 2.
The figure shows that the coating window is bounded by
three modes of failure:
(1) When the coated layer is thicker than the thinnest that
can be produced at a fixed gap and substrate speed, i.e.
t > tmin in Fig. 2, too great a vacuum at the upstream
free surface causes liquid to be drawn along the die surface into the vacuum chamber. This diversion of liquid
destroys premetering.
(2) Too little vacuum at the upstream free surface leaves
the net viscous drag force on the upstream part of the
Pvac Pvac
Patm
(1)
(3)
Patm
Pvac
COATING
Pvac
WINDOW
Patm
Low flow limit
(2)
H0 /t0
H0 /tmin
Pvac
Patm
Ca
H0 /t
Fig. 2. Coating window of a slot coating process in the plane of vacuum
Pvac vs. gap-to-thickness ratio H0 /t, at a fixed capillary number Ca. The
boundaries of the window are set by different bead break-up mechanisms.
tmin represents the minimum film thickness that can be deposited onto
the substrate at a given capillary number.
bead unbalanced by the pressure gradient that is imposed by capillary pressure forces in the menisci upstream and downstream and the difference in external
pressure on those menisci (i.e. vacuum). In consequence
the upstream meniscus shifts toward the feed slot until
the bead drastically rearranges into a three-dimensional
form that delivers separate rivulets to the substrate. Between the rivulets are dry lanes that extend upstream
through the bead. Along those lanes air is sucked into
the vacuum chamber. It is in this regime that, at given
vacuum (ambient pressure downstream minus air pressure exerted on upstream meniscus), there is a lower
limit to the thickness of continuous liquid layer that can
be coated from a downstream gap of specified clearance.
As Fig. 2 shows, the limit can be lowered by applying greater vacuum and thereby shifting the upstream
meniscus away from the edge of the feed slot.
(3) At given substrate speed, too low a flow rate per unit
width from the slot causes the downstream meniscus to
curve so much that it cannot bridge the gap clearance
H0 . Consequently, the meniscus becomes progressively
more three-dimensional, alternate parts of it invading
the gap until the bead takes a form that delivers separate rivulets or chains of droplets to the substrate moving past. This transition from a continuous coated liquid
layer is what is called here the low-flow limit: the minimum thickness of liquid that can be deposited from a
gap of specified clearance at a given substrate speed.
And, as Fig. 2 makes plain, it is independent of the vacuum applied, given that the vacuum is great enough to
draw the upstream meniscus away from the feed slot.
The outcome is the same when at a given flow rate per
unit width from the slot, the substrate speed is too high.
In this case, the low-flow limit is sometimes referred to
as the high-speed limit (see [4]). The outcome is essentially the same when at a given flow rate per unit width
from the slot and a given substrate speed, the clearance
of the downstream gap is too great. In this case, the
low-flow limit is refered to as the wide-gap limit: the
maximum gap from which a given thickness of liquid
coating can be deposited on a substrate moving at specified speed.
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
139
feed slot
upstream
downstream
DCL
DOWNSTREAM
SLOT DIE
SCL
H0
R min
liquid
t min
coa ted la yer
V
substrate
Fig. 3. Detail of the downstream meniscus configuration. The low-flow limit occurs when the free surface is as curved as it can be and still meet the die.
The threshold of the low-flow limit is well described
by the viscocapillary flow model, provided viscous stress
at the downstream meniscus is not excessive and the inertia of the liquid is not appreciable. In this model, the
streamlines are rectilinear and parallel, or nearly so, in
most of the flow; the upstream meniscus is part of a circular cylinder and the downstream meniscus nearly so but
governed by the Landau–Levich relation, which is strictly
valid only for flow at low capillary number, Ca ≡ µV/σ
(viscosity times substrate velocity divided by surface tension) and Reynolds number, Re ≡ ρq/µ (density times
flow rate per unit width of slot divided by viscosity). According to the viscocapillary model developed by Higgins
and Scriven [15], given the substrate speed, the minimum
flow rate per unit width—and hence the minimum liquid
layer thickness—is set by the greatest adverse pressure
gradient that can be created in the downstream part of the
gap. That arises when the upstream meniscus is hemicylindrical and situated just upstream of the feed slot, so that
the pressure jump there is the highest it can be, and the
downstream meniscus is as curved as it can possibly be and
still connect the free surface of the thin coated layer to the
surface of the die lip that defines the downstream gap, as
illustrated in Fig. 3. The maximum pressure difference is
given by:
Pmax =
2σ
,
H0 − t
(1)
t is the coated film thickness. Ruschak [24] extended the
work of Landau and Levich [17] and showed that the pressure drop across a downstream meniscus in a slot coating
bead can be related to the film thickness t, the web speed V
and liquid properties:
σ
P = 1.34Ca2/3 .
(2)
t
The minimum liquid layer thickness at a given capillary
number Ca can be obtained by combining the two previous
equations:
1
tmin
=
;
H0
1 + 1.49Ca−2/3
Ca 1, Re 1.
(3)
Above the critical gap-to-thickness ratio, two-dimensional
steady flow cannot exist, according to the viscocapillary
flow model. Moreover, the formula indicates that the minimum thickness of liquid that can be coated from a downstream gap of specified clearance is greater, the higher the
capillary number. Thus, for example, raising the production
speed makes the minimum liquid layer thicker, which might
be accommodated by adding solvent, which in turn would
heighten the demand on the drying operation downstream
of the coater.
Experiments by Sartor [25] and Carvalho and Kheshgi
[4] with Newtonian liquids accorded with the viscocapillary model at capillary numbers below unity but indicated,
as expected, that it becomes less and less useful as capillary number and Reynolds number rise beyond unity. High
capillary number (Ca ≈ 5) and Reynolds number (Re ≈
3) occur in higher-speed coating operation. However, the
film profile equation of Kheshgi [16], which expresses the
viscocapillary model augmented to take partial account of
liquid inertia, represents the data to somewhat higher values of these parameters, where it, too, fails and beyond
which only the full Navier-Stokes system accords with the
data.
In the experiments of Lee and Liu [19] as in those of Gutoff and Kendrick [14] to determine a “low-flow limit”, no
vacuum was applied at the upstream meniscus. The results
therefore correspond to Point t0 in Fig. 2, which is obviously a thickness greater than the low-flow limit tmin , which
requires vacuum.
Solving the Navier-Stokes system for two-dimensional
viscous flow with free surfaces, Carvalho and Kheshgi [4]
confirmed the viscocapillary model of the low-flow limit at
low capillary number and low Reynolds number and, more
importantly, found that the limit, instead of continuing to
140
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
rise with these parameters, begins to fall when they reach
higher values, in accord with their experiments.
Most past analyses of slot coating flow have dealt solely
with Newtonian liquids. However, the liquids coated in
practice are polymer solutions or colloidal suspensions, or
both. In general, polymer solutions may be shear-thinning,
extension-thickening, and develop viscoelastic stresses
when sheared or extended. These liquids can depart substantially from Newtonian behavior at the strain rates they
suffer in the coating flow. In the particular case of the
low-flow limit, such rheological effects can change the
forces that operate near and at the downstream meniscus of
the coating bead and thereby alter the limit.
Dontula [10] analyzed experimentally the flow of polymer solutions in several coating processes. Model aqueous
solutions with nearly constant viscosity and adjustable elasticity were developed to study the role of elasticity in coating and other free surface flows (see also [9]). These liquids
were prepared by adding small amounts of a high molecular weight polymer (poly-ethylene oxide—PEO) to a more
concentrated aqueous solution of the same polymer but of a
much lower molecular weight (polyethylene glycol—PEG).
The experiments showed that the free surface flow in the
coating bead can be drastically changed even when only
minute amounts of the high molecular polymers were added
to the coating liquid. The elasticity appears to arise from
the strongly extensional nature of the flows at the downstream free surface of the coating bead. However, at the low
concentrations of the high molecular weight polymer, the
elastic moduli of the polymeric solutions were so small that
they could not be discerned from small-amplitude oscillatory
tests. The only measurable manifestation of non-Newtonian
behavior was the rise of the apparent extensional viscosity.
Ning et al. [21] studied experimentally the effect of polymer additives on slot coating. However, like Lee and Liu
[19], they used no vacuum (reduced pressure) on the upstream meniscus and so the “coatability limit” they measured is not, as claimed, the low-flow limit of the process.
In this work, the low-flow limit in slot coating of certain
dilute aqueous polymeric solutions was examined by both
experiments and theory. With a benchtop slot coating apparatus, a coating gap and substrate speed were set. A coating
bead was established by feeding liquid to the slot die at high
enough volumetric flow rate per unit width being coated that
the deposited layer was half the thickness of the gap and the
bead was easy to establish. Then the flow rate was slowly
reduced until the coated layer broke into alternating stripes
of liquid and dry lanes. At each such operating condition,
the positions of the downstream and upstream menisci were
observed before and after the break in order to determine
the breakup mechanism. Aqueous solutions, like the ones
developed by Dontula et al. [9], of low molecular weight
polyethylene glycol to adjust viscosity, and high molecular
weight polyethylene oxide, to adjust elasticity, were used in
order to evaluate the effects of non-Newtonian behavior. The
theoretical analysis consisted of solving the momentum and
continuity equation system for steady two-dimensional flow
with free surfaces, including an algebraic non-Newtonian
constitutive equation that relates stress to the rate-of-strain
and relative-rate-of-rotation tensors. Generalized Newtonian
models of this class cannot represent viscoelastic stresses
per se but are perhaps the simplest ones that may capture
the different ways that dilute polymer molecules behave in
extension-dominated and shear-dominated flow zones. The
system of differential equations, boundary conditions and
constitutive equation was solved by the Galerkin/finite element method. At each operating condition (set by the liquid properties as well as coating gap and web speed), a
sequence of states was found by Newton iterations initialized by first-order, pseudo-arc-length continuation, i.e. a solution path in parameter space was constructed as the flow
rate to the die was diminished. The theoretically determined
low-flow limit was defined by the parameter values at which
the angle between the downstream free surface and the
downstream die lip became less than 10◦ .
2. Experimental analysis
2.1. Experimental set-up and procedure
The experimental set-up is sketched in Fig. 4. The coating liquid was fed to the coating die by a progressive cavity
pump (model C4015ESSQ3SAA, Moyno Industrial Products, Springfield, OH). The flow rate was controlled by the
pump speed and measured with a Coriolis flow meter (model
MFC 100/MFS 3000, KROHNE Americ Inc, Peabody, MA).
The precision coating die, 10 cm wide, was assembled with
shims such that the height of the slot was 125 ␮m. A vacuum
chamber of approximately 300 cm3 was mounted upstream
the coating die. Two vacuum pumps (model VFC084P-5T,
Fuji Electric Co, Tokyo, Japan) in series were used in order to achieve high enough vacuum (low enough pressure)
to maintain the coating bead at the highest substrate speeds.
The set-up was able to maintain a pressure of approximately
8 kPa below atmospheric. The coated film thickness was
uniform across the width of the die; hence it was calculated simply by dividing the flow rate by the coating width
and surface speed of the cylinder being coated. The die was
mounted on an adjustable plate, such that the coating gap,
i.e. the distance between the die and the cylinder surface,
could be easily changed and made uniform. In most of the
experiments, the coating gap was fixed at 100 ␮m. The liquid was coated directly onto the glass cylinder, or roll. Its
surface speed could be controlled from 0.2 up to 2.8 m/s.
The coated liquid, except a trace residue, was removed from
the roll surface by a scraper blade and recycled. A video
camera (model DXC-750, Sony Inc., Tokyo, Japan) with microscope lens (Magnazoom 6000 with 0.5×adapter, Navitar,
Rochester, NY) was focused on a right-angle mirror mounted
inside the open end of the roll in order to visualize the flow in
the coating bead through the glass wall, as shown in Fig. 5.
The coating liquids used in the experiments were solutions
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
141
FILM THICKNESS
GLASS ROLL
SLOT COATER
Pressure sensor
SCRAPER 1
VACUUM
BOX
MASS FLOWMETER
LIQUID PAN
SCRAPER 2
PROGRESSIVE
CAVITY PUMP
LIQUID TRAP
HAND-SET
VALVE
VACUUM PUMP
Fig. 4. Schematic of the experimental slot coating apparatus.
of polyethylene glycol (molecular weight 8 × 103 g/mol)
and polyethylene oxide (molecular weight 4 × 106 g/mol).
Three concentration of PEG were used, 20, 25, and 30%
by weight, and to each, small amounts of PEO were added
to give them slight to moderate viscoelastic character. Four
different concentrations of PEO were used: 0, 0.005, 0.01,
and 0.05% by weight. The rheological characterization of
the solutions is described in the next subsection. The equilibrium surface tension of the solutions was measured with
a Kruss Drop Analysis System DSA10, and were all in the
range of 58–61 mN/m.
The minimum liquid layer thickness that could be coated
at each speed of the cylinder was found by the following
procedure:
(1) The pump speed was set to deliver a relatively thick
coated layer, i.e. approximately 50 ␮m—half of the coating gap. Enough vacuum was applied to the upstream
free surface, or meniscus, to establish the coating bead;
in some cases this was aided by drawing the tip of a
slender wooden stick across the gap exit (a common expedient in practical coating).
(2) The pump speed was lowered in small steps. After less
than 10 s, steady state was reached and the positions of
the dynamic contact line and the downstream static contact line were observed. If the upstream meniscus had
moved toward the feed slot, the vacuum was raised in
order to pull the meniscus back to about its former location. The flow rate was lowered until the downstream
Fig. 5. Photograph of the experimental set-up with camera and microscope lens.
142
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
meniscus invaded the coating bead. Details of how the
bead broke up are described in following subsections.
After this happened, one or more lanes of uncoated roll
surface appeared and no manipulation with the pointed
wooden stick at the gap exit would re-establish a continuous, uniform coating. This operating condition was
considered to be the onset of the low-flow limit. The
minimum layer thickness that could be coated was calculated from the flow rate and the surface speed of the
roll.
2.2. Rheological characterization of the solutions
The coating liquids used in the experiments were solutions
of polyethylene glycol (molecular weight 8 × 103 g/mol)
and polyethylene oxide (molecular weight 4 × 106 g/mol).
These aqueous polymer solutions were proposed by Dontula et al. [9] as model liquids to study the role of elasticity in coating and other free surface flows. Similar solutions
were used by Christanti and Walker [5] and Cooper-White
et al. [6] to study jet break up and drop formation dynamics. The solutions were transparent and behaved similarly
to Boger liquids. Their rheological properties depended on
concentration. Dontula et al. [9] prepared different solutions
of PEG and PEO in water in their study. The concentration of PEG they have tested varied from 16 to 43 wt.%,
and that of PEO, from 0 to 0.1 wt.%. They characterized
the rheological properties of the solutions in shear and in
approximately extensional flows. The elastic modulus G
could be reliably measured only in the more viscous solutions (larger PEG concentration and higher concentration of
PEO). The apparent extensional viscosity of the solutions
was measured in an opposed nozzle device. The higher concentration solutions displayed an extension-thickening behavior that presumably came from the stretching of polymer
coils.
In the present work, 12 different solutions with different
concentrations of PEG and PEO were tested; the concentrations were lower than those reported by Dontula et al.
[9]. The elastic modulus G of the solutions containing PEO
could not be measured in small amplitude oscillatory tests
because of the high uncertainties at small torques. The shear
viscosity was measured first in a RFS-II rheometer (Rheometrics Inc., Piscataway, NJ) with a cone-and-plate fixture
of diameter of 50 mm and angle of 0.04 rad. The shear viscosity of all solutions were virtually independent of shear
rate. In order to determine the inherent viscosity of the solutions tested as a function of the PEO concentration with
higher accuracy, the shear viscosity was also measured using glass capillary viscosimeter (Cannon-Fenske). Table 1
presents the viscosity of the solutions at different PEO concentration (from 0 to 0.1 wt.%) in 20 and 30 wt.% PEG solutions. The variation of the inherent viscosity, defined as
(η0 − ηs )/ηs with PEO concentration in presented in Fig. 6.
It is independent of the PEG concentration and varies linearly with the high molecular weight polymer (PEO) concentration. The intrinsic viscosity, i.e. the slope of the line,
is approximately [η] ≈ 0.8 m3 /kg. At equilibrium, the polymer coils start to overlap when the reduced concentration,
defined as c∗ = c × [η], is approximately c∗ = 1 (see [20]).
The maximum reduced concentration of the solutions tested
here was below this limit and consequently all the test liquids were in the dilute regime.
The relaxation time of the solutions were estimated by
taking λ ≡ ηp /G, where ηp is the polymer contribution for
the viscosity (ηp = η0 − ηs ) and G is the elastic modulus of
the liquid. It is a function of the polymer concentration c and
its molecular weight Mw (G ≡ (c/Mw )RT). The estimated
relaxation time of each solution is also presented in Table 1.
As expected in the dilute regime, the relaxation time is independent of the PEO concentration. The estimated relaxation
time was λ ≈ 0.02 and ≈ 0.06 s for the 20 and 30 wt.%
PEG solutions, respectively.
The solutions’ extensional response was probed with an
opposed nozzle device (RFX, Rheometrics Inc, Piscataway,
NJ). Fig. 7 presents the apparent extensional viscosity of all
the solutions tested. Several nozzle diameters were used to
cover the range of nominal extensional rates reported, from
100 to 4000 s−1 . The breaks between sets of points associated with each solution are where the nozzle diameter was
changed. This sensitivity of the measurements to the nozzle
diameters has been reported before (see [8]). The apparent
extensional viscosity could not be measured at lower apparent extension rate because the torques generated were below
Table 1
Estimated relaxation time of the solutions at different PEO concentration
PEG (wt.%)
PEO (wt.%)
c (kg/m3 )
ηs (mPa s)
η0 (mPa s)
(η0 − ηs )/ηs
G (Pa)
λ (s)
20
0
0.01
0.03
0.05
0.1
0
0.103
0.310
0.517
1.035
17.07
17.07
17.07
17.07
17.07
17.07
18.86
21.83
25.13
30.96
0
0.105
0.278
0.472
0.813
0.00
0.06
0.19
0.32
0.64
0.000
0.028
0.025
0.025
0.022
30
0
0.01
0.03
0.05
0.1
0
0.105
0.315
0.526
1.054
44.48
44.48
44.48
44.48
44.48
44.48
51.56
57.04
63.78
84.09
0
0.159
0.282
0.434
0.891
0.00
0.06
0.19
0.32
0.64
0.000
0.108
0.064
0.060
0.061
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
143
1
Inherent viscosity, (η0−ηs)/η s
0.9
Water+PEG(20%)+PEO
Water+PEG(30%)+PEO
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
3
PEO concentration, kg/m
Fig. 6. Inherent viscosity (η0 − ηs )/ηs as a function of PEO concentration. The intrinsic viscosity is [η] ≈ 0.8 m3 /kg.
the range of the sensor. In the range at which the measurements were made, the apparent extensional viscosity of all
the solutions tested was nearly independent of apparent extension rate. Presumably the apparent extensional viscosity
had reached the constant value associated with high enough
extension rate that all the coils are stretched out [22]. From
the estimated relaxation time of each liquid tested, the different PEO solutions on 20 wt.% PEG in water should show
an extensional thickenning behavior at a deformation rate of
γ̇ ≈ 1/λ ≈ 50 s−1 , which is lower than the minimum extensional rate at which reliable measurements could be made.
The three solutions lacking PEO appeared to be Newtonian in that the ratio of their apparent extensional to shear
viscosity was approximately 4. This ratio is often referred to
as a Trouton ratio (although it is not Trouton’s). The higher
the concentration of PEO, the greater was the almost constant apparent extensional viscosity. In the case of the solution with 30% PEG and 0.05% PEO, the ratio of apparent
extensional to shear viscosity was approximately 15.
2.3. Results
Flow visualization through the glass wall of the rotating cylinder used as substrate revealed how the upstream
and downstream menisci of the coating bead and their contact lines located after each change of operating conditions,
as illustrated in Fig. 8, a sequence of photographs as the
flow rate falls with no vacuum applied to the upstream free
meniscus. In every case shown in Fig. 8, the downstream
contact line pinned at the corner (radius of about 25 ␮m) of
the downstream land but the upstream contact line located
in response to the particular operating conditions. At large
enough flow rates, the flow was two dimensional and both
contact lines were straight—Fig. 8(a). As the flow rate fell
at constant roll speed (Ca = 0.12), the upstream contact
line moved toward the feed slot. At a certain operating condition, that contact line reached the feed slot and became
three-dimensional; a “V” shaped air pocket formed at the
dynamic contact line—Fig 8(b). At even lower flow rate, the
downstream contact line invaded the gap and the bead broke,
i.e. it became three-dimensional and delivered stripes rather
than continuous liquid layer—Fig. 8(d). However, as noted
above, this was not the way the bead failed at the low-flow
limit. This type of coating failure could be avoided by raising the vacuum, i.e. reducing the pressure on the upstream
meniscus, and thereby pulling it away from the feed slot.
The response of the contact lines to the vacuum applied is
shown in Fig. 9.
At the operating conditions at which the deposited liquid
film thickness was approximately half of the gap between
the die and the rotating cylinder, the downstream contact line
located at the corner of the downstream die lip, as shown
in Fig. 8. Previous experiments [25] have shown the downstream contact line located on the die shoulder; however in
those experiments the coating thicknesses were larger than
the ones explored here and the substrate speed were lower.
Close to the onset of the low-flow limit, in contrast, the
downstream meniscus receded into the coating gap, i.e. toward the feed slot, locally at certain locations across the
width of the coating bead. The sequence of photos of the
coating bead in Fig. 10 shows this mode of failure. The
capillary number was Ca = 0.084, and the vacuum applied to the upstream free surface was Pvac H0 /µV = 180.
When the coated layer was thick (low H0 /t), the flow was
two-dimensional and the contact lines were straight. As the
flow rate was reduced, the forces of the two-dimensional
flow evidently could not balance close to the downstream
meniscus, for the flow there became three-dimensional at
H0 /t = 4.5 (see Fig. 10(c)). When the flow rate was reduced even further, e.g. H0 /t = 5.45, air fingers formed at
144
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
App. Extensional Viscosity, cP
1000
100
(a)- water + PEG(20%)+
PEO(0.05%)
PEO(0.01%)
PEO(0.005%)
PEO(0%)
10
10
App. Extensional Viscosity, cP
1000
100
1000
10000
App. Extension Rate, s-1
100
PEO(0,05%)
(b)- water + PEG(25%)+
PEO(0,01%)
PEO(0,005%)
PEO(0%)
10
10
App. Extensional Viscosity, cP
1000
100
1000
App. Extension Rate, s
10000
-1
100
PEO(0,05%)
(c)- water + PEG(30%)+
PEO(0,01%)
PEO(0,005%)
PEO(0%)
10
10
100
1000
App.Extension Rate,s
10000
-1
Fig. 7. Apparent extensional viscosity as a function of apparent extension rate of the polymeric solutions used. (a) 20 wt.% PEG; (b) 25 wt.% PEG and
(c) 30 wt.% PEG.
the downstream lip and penetrated all the way through the
bead to the vacuum chamber, so that one or more dry stripes
appeared on the cylinder (see Fig. 10(d)). It is important to
notice that the upstream free surface remained well upstream
of the feed slot, for that proves that mechanism that caused
the low-flow limit is related to the movement of the downstream meniscus. Such a mechanism of the low-flow limit
accords with the theoretical models of [24,15], the more refined one of [12], and that of [4]. It is clear that the bead
breakup seen in earlier experiments in which no vacuum was
applied to the upstream meniscus was not the bead breakup
at the low-flow limit in slot coating.
The conditions at the onset of the low-flow limit of 20%
PEG solutions measured experimentally are summarized in
Fig. 11, which also records the critical gap-to-thickness ratio versus capillary number at the onset of the low-flow limit
that is predicted by the viscocapillary model. The critical
condition found with the solution lacking PEO agrees well
with the viscocapillary model based on Newtonian behavior,
as expected because the low molecular weight polymer solution is a Newtonian liquid and the experiments were performed at low enough capillary number that the viscocapillary model is accurate. At a fixed capillary number, as the
concentration of PEO rises and consequently the viscoelasticity of the liquid does too, the minimum thickness that can
be coated rises. At Ca = 0.2, the minimum layer thickness
that could be coated with the 0.05% PEO solution was close
to twice of that which could be coated with the solution
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
145
Fig. 8. View through the glass roll of the coating bead as the flow rate is reduced. The roll speed was fixed (Ca = 0.12) and no vacuum was applied to the
upstream free surface. The cylinder was moving from bottom to top of each photo. As the film thickness falls, the upstream free surface moves towards
the feed slot and becomes three-dimensional. As the flow rate is decreased even further, the “V” pattern grows until the coating bead breaks (Frame d).
lacking PEO. The coating window of the process was smaller
the higher the extensional viscosity of the liquid used.
The low-flow limits of the solutions with 25 and 30 wt.%
PEG are shown in Fig. 12. The trend is similar to that in
Fig. 11, i.e. the addition of small quantities of PEO made
the minimum thickness larger. A minor difference is that the
critical condition found with the solution lacking PEO is not
as close to the predictions of the viscocapillary model with
Newtonian behavior.
Because the flow near the downstream meniscus is dominated by extensional deformation, the ratio of the viscous
to the surface tension force is better described by a capillary
number based on the apparent extensional viscosity of the
coating solutions, viz. Cau ≈ ηu V/3σ. The data of Fig. 11
are reploted in Fig. 13 as a function of this modified capillary number. The apparent extensional viscosity used in the
definition of the modified capillary number is the one measured at the highest extensional rate at which the data was
taken, e.g. ˙ ≈ 4000 s−1 . When the extensional thickening
of the liquid is taken into account in this way in the definition of the capillary number, all the data lie approximately
on the same curve.
The shape of the downstream meniscus as it invades the
coating bead at the critical conditions of the onset of the
low-flow limit depends on the forces acting on that meniscus. As the liquid becomes more viscoelastic, i.e. as the
Fig. 9. Coating bead configuration at H0 /t = 2, Ca = 0.12 and increasing level of vacuum applied to the upstream free surface. With no vacuum,
Pvac = 0, the bead is not continuous (Frame a). At Pvac = 98.5, an uniform coating bead is formed (Frame d).
146
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
Fig. 10. Position of the downstream contact line as flow rate is reduced. The low-flow limit occurred at H0 /t = 5.4 (Frame d). The flow conditions were
Ca = 0.084 and Pvac H0 /µV = 180.
Capillary Number, Ca = ηοV/σ
10
PEO(0%)
water + PEG(20%) +
PEO(0.005%)
PEO(0.01%)
PEO(0.05%)
1
UNSTABLE
Viscocapilllary Model
0.1
STABLE
0.01
0
2
4
6
8
10
12
Gap / Film Thickness, H0 / t
Fig. 11. Onset of the low-flow limit as a function of capillary number. Twenty weight percent PEG solutions.
concentration of high molecular weight PEO rises, the “V”
shaped meniscus profile becomes more acute, as illustrated
in Fig. 14.
3. Theoretical analysis
The flow in the coating bead was described by the complete two-dimensional, steady-state mass and momentum
conservation equations coupled with an algebraic constitutive equation that approximates the non-Newtonian behavior
of the liquid. This section describes the formulation of the
theoretical model, how its equations were solved, and the
predictions solutions yielded.
Carvalho and Kheshgi [4] established that at low capillary
and Reynolds numbers the flow upstream of the feed slot
does not affect the critical operating conditions at the onset
of the low-flow limit. Because the analysis presented here
focuses on the effect of non-Newtonian behavior of the liquid
at low capillary number and vanishing Reynolds number,
the flow domain where the governing equations are solved
could be restricted to the region close to the downstream free
surface, as sketched in Fig. 15. H0 is the gap between the
die and the moving substrate. The synthetic inflow boundary
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
147
Fig. 12. Onset of the low-flow limit as a function of capillary number. (a) 25 and (b) 30 wt.% PEG solutions.
3)V
10
Capillary Number*, CaU = (
0U
PEO(0%)
PEO(0.005%)
water+PEG(20%)+
PEO(0.01%)
PEO(0.05%)
1
Viscocapillary Model
0.1
0.01
0
2
4
6
8
10
12
Gap / Film Thickness, H0 / t
Fig. 13. Low-flow limit of 20 wt.% PEG solutions as a function of the modified capillary number, defined in terms of the apparent extensional viscosity.
148
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
Fig. 14. Shape of the three-dimensional downstream contact line at the onset of the low-flow limit for (a) Newtonian liquid (0 wt. % PEO) and (b)
extensional-thickening solution (0.05 wt.% PEO).
was placed at 5H0 upstream of the static contact line. The
outflow plane was placed 15H0 downstream of the static
contact line.
In order to validate the predictions obtained with this
restricted domain, the predicted low-flow limit for Newtonian liquids was compared with Carvalho and Kheshgi’s
[4] results of solving the governing equations in a complete
two-dimensional cross-section of the coating bead.
3.1. Governing equations and boundary conditions
The velocity and pressure field of Stokes flow are governed by the momentum and continuity equations:
∇ · T = 0; ∇ · v = 0.
(4)
¯¯
¯
¯
The constitutive equation that relates the stress T and the
¯¯
liquid deformation is presented in the following section.
The boundary conditions that were adopted are the following:
(1) Inflow: Couette–Poiseuille velocity profile.
6q
y
y 2
u=−
; v = 0.
−
H0
H0
H0
(5)
Here q is the flow rate per unit width, an input parameter,
and H0 is the gap between the die and the substrate.
(2) Moving substrate: no-slip, no-penetration.
u = Vw ;
v = 0,
(6)
where, Vw is the substrate velocity.
(3) Outflow: fully developed flow.
n · ∇v = 0.
¯
¯
(7)
(4) Free surface: kinematic condition and force balance.
n · v = 0;
¯ ¯
1 dt
n·T =
¯ − npamb .
Ca ds ¯
¯ ¯¯
(8)
here t and n are the unit tangent and unit normal vector
¯ surface, and p
along¯ the free
amb is the ambient pressure.
(5) Static contact line: pinned at the corner of the die lip.
xscl = xcorner .
¯
¯
(6) Die land: no-slip, no-penetration.
v=0
¯
(9)
(10)
3.2. Constitutive equation
Microstructured liquids can depart substantially from
Newtonian liquid behavior in complex flows. Shear-thinning
and extension-thickening may vary with the intensity of
deformation rate, and viscoelastic stresses may arise. Moreover, polymer molecules in solution behave quite differently
in flow regions where the liquid is persistently stretched in
the direction in which the molecules are aligned, as compared to regions where the molecules rotate with respect to
the rate-of-strain field. Some differential constitutive equations can qualitatively describe how polymer molecules
respond to flow kinematics. The solution of viscoelastic
free surface flows using differential constitutive models was
presented recently by Pasquali and Scriven [23] and Lee
et al. [18].
Algebraic models that relate stress to the rate-of-strain
tensor and relative-rate-of-rotation tensor cannot account
for viscoelastic stresses per se but are perhaps the simplest way of trying to capture the distinctive behavior of
Fig. 15. Sketch of flow domain.
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
polymer molecules in extension-dominated as compared to
shear-dominated flow zones. Using this type of constitutive
equation can enhance understanding of the effect of soluble
polymers in coating liquids. In the particular case of free
surface flows of low concentration solutions of high molecular weight polymers, the shear thinning and extensional
thickening effects appear to have more consequences than
viscoelastic behavior [27].
The rate-of-strain and the relative-rate-of-rotation are:
D ≡ 21 [∇v + (∇v)T ] and W̄ ≡ W − Ω,
¯¯
¯¯
¯
¯
¯¯
¯¯
T
where W = 1/2[∇v − (∇v) ] is the vorticity tensor, and
¯
Ω is the¯¯ tensor that¯ represents
the angular velocity of the
¯
¯
principal
direction of the rate-of-strain tensor following the
liquid particle motion. Astarita [1] defined a flow classification index to measure the degree to which a fluid particle
avoids stretching; the index R is based on the relative rate
of rotation W̄ and rate of deformation D tensors:
¯¯
¯¯
2
tr(W̄ )
R ≡ − ¯¯ 2 .
(11)
tr(D )
¯¯
The flow classification index takes the value, R = 0, in pure
extension and the value, R = 1, in shear flows. Moreover,
as the flow approaches a rigid body motion, i.e. as D → 0,
¯¯
R approaches infinity.
Schunk and Scriven [27] adopted the flow classification
index defined by Astarita [1] and proposed a generalized Newtonian model in which the liquid viscosity is an
arithmetic average of the shear viscosity and the uniaxial
extensional viscosity, the latter being represented by a constitutive equation of the same functional form as Carreau’s
for shear viscosity. The constitutive relation put forward by
Thompson et al. [29] follows the same idea, but extends the
capability of this type of model, for it is able to describe
shear thinning and normal stress differences in simple shear
flow and extensional thickening in pure extensional flow
simultaneously. For the stress tensor it gives
T = −pI + α1 D + α3 D2 + α4 [D · W̄ − W̄ · D].
(12)
¯¯
¯¯
¯¯
¯¯
¯¯ ¯¯
¯¯ ¯¯
This is the most general way of writing a stress tensor that
is symmetric as a function of the rate-of-strain and relative
rate-of-rotation tensor, and can fit independently measured
material functions of shear viscosity and extensional viscosity. The coefficients in (12) depend on not only those
material functions of the liquid, but also the first and second
normal stress coefficients. Here, a simplified version of this
equation, a generalized Newtonian model, is used:
T = −pI + α1 D.
(13)
¯¯
¯¯
¯¯
The coefficient α1 is a function of the shear viscosity ηs , a
viscosity in extension ηu , defined as ηu ≡ Txx − Tyy /3˙ in
uniaxial extension flow, and the flow classification index R:
(1−R)
.
α1 = 2ηR
s ηu
(14)
149
Each viscosity function depends on the deformation rate; in
the framework of this class of models, the functions describing these dependencies are found empirically. Although
Eq. (13) does not account for viscoelastic behavior per
se, it is able to describe the effect of extension-thickening
common to high molecular weight polymeric solutions.
3.3. Solution method
Because of the free surfaces, the flow domain at each set
of parameter values was unknown a priori. To solve this
free boundary problem by means of standard techniques
for boundary value problems, the set of differential equations and boundary conditions posed in the unknown domain
Eq. (4) had to be transformed to an equivalent set defined in
a known reference domain. This transformation was made
by a mapping x = x(ξ ) that connects the two domains. The
¯
¯domain
¯
unknown physical
was parameterized by the position vector x, and the reference domain by ξ . The mapping
¯ the one described by De Santos
¯ [7]. The inused here was
verse of the mapping that minimizes the functional is governed by a pair of elliptic differential equations identical to
those encountered in diffusional transport with variable diffusion coefficients. The coordinates potentials ξ and η of the
reference domain satisfied
∇ · (Dξ ∇ξ) = 0;
∇ · (Dη ∇η) = 0.
(15)
Dξ and Dη are diffusion-like coefficients used to control gradients in coordinate potentials, and thereby the spacing between curves of constant ξ on the one hand and of constant
η on the other that make up the sides of the elements that
were employed; they were quadrilateral elements. Boundary
conditions were needed in order to solve the second-order
partial differential Eq. (15). The solid walls and synthetic
inlet and outlet planes were described by functions of the
coordinates and along them stretching functions were used
to distribute the terminii of the coordinate curves selected to
serve as element sides. The free boundary (gas–liquid interface) required enlarging the system of governing equations
with the kinematic condition, viz. Eq. (8). The discrete version of the mapping equations is generally referred to as
mesh generation equations.
The system of governing equations was solved by
Galerkin’s method with quadrilateral finite elements. The
velocity and node positions were represented in terms of
biquadratic basis functions φj , and the pressure in terms of
linear discontinuous basis functions χj :
u = nj=1 Uj φj ; v = nj=1 Vj φj ; p = m
j=1 Pj χj ;
n
n
(16)
x = j=1 Xj φj ; y = j=1 Yj φj
because the stress tensor depended on the second derivative
of the velocity field (through the definition of the index R)
and that derivative is discontinuous across the sides of biquadractic elements, an additional variable L was introduced
¯¯
to represent the velocity gradient as a continuous
function,
150
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
i.e. L = ∇v. It was represented in terms of bilinear basis
¯¯
functions
ψ¯j :
Lux =
Lvx =
p
LUXj ψj ;
Luy =
j=1 LVXj ψj ;
Lvy =
pj=1
p
pj=1
LUY j ψj ;
j=1 LVY j ψj .
(17)
Thus, the system of partial differential equations, and boundary conditions was reduced to a set of simultaneous algebraic
equations for the coefficients of the basis functions of all
the fields. This set was non-linear and sparse. It was solved
by Newton’s method. In order to improve the initial guess
at each new set of operating conditions a pseudo-arc-length
continuation method, as described by Bolstad and Keller [3],
was used. The first successful free surface flow was computed using a fixed boundary flow field with slippery surface in place of the free boundary as the initial condition for
Newton’s method.
The domain was divided into 256 graded quadrilateral
elements, with 6400 unknowns; tests showed that using more
elements produced changes in the values of the coefficients
no greater than 1%. Fig. 16 shows detail of the mesh near
the free surface.
3.4. Results
Solutions were computed at capillary numbers from Ca =
0.11 up to 1.5; these values cover the range from where
capillary pressure is dominant in at least part of the flow
to where it is generally comparable in magnitude to viscous stress. The operating conditions at the low-flow limit
were determined theoretically by following a solution path
constructed at a fixed capillary number (e.g. fixed substrate
velocity, liquid viscosity and surface tension) by lowering
the flow rate, and consequently the thickness of the coated
liquid layer.
3.4.1. Newtonian liquid: flow kinematics and low-flow limit
The flow field as a function of the layer thickness and
hence flow rate is shown in Fig. 17 at capillary number Ca =
0.3. As the thickness of the deposited liquid falls, a recirculation attached to the free surface appears, the meniscus
becomes more curved, and the inclination angle of the free
surface with the downstream die land diminishes. Fig. 18
displays the computed static contact angle at different capillary numbers and gap-to-thickness ratios. At a fixed flow
rate, i.e. fixed gap-to-thickness ratio, the static contact angle rises as the capillary number falls, as predicted first by
Saito and Scriven [26] and later by Sartor [25], Gates and
Scriven [12] and Carvalho and Kheshgi [4]. At low capillary
number, surface tension force (capillary pressure) is strong
and the meniscus does not need to curve as much in order to
provide the adverse pressure gradient needed to match the
flow in the gap to the rate at which liquid is fed.
The onset of the low-flow limit at a given capillary number was determined by recording the gap-to-thickness ratio
at which the static contact angle reached θ = 100 , as indicated in Fig. 18. The critical conditions is graphed as a
function of the capillary number in Fig. 19. The graph also
shows the critical conditions predicted by the viscocapillary
model and those predicted by solving the governing equations of two-dimensional flow over the entire length of the
coating bead, as reported by Carvalho and Kheshgi [4]. At
high capillary number (Ca > 0.05), the viscocapillary model
underpredicts the critical gap-to-thickness ratio, i.e. overpredicts the minimum film thickness. Important to what follow
is the fact that the critical conditions found in the present
Fig. 16. Detail of the mesh near the free surface.
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
151
Fig. 17. Evolution of the streamlines as the flow rate, i.e. film thickness, falls. Ca = 0.3.
work by solving the governing equations only in a region
close to the downstream free surface of a slot coating bead
are virtually the same as those predicted by Carvalho and
Kheshgi [4], and agree well with the experiments reported
in the previous section and those presented by Carvalho and
Kheshgi [4].
Fig. 20 shows how the second invariant of the rate-of-strain
tensor evolve as a function of gap-to-thickness ratio at
Ca = 0.3. The deformation rate is plotted in units of web
speed over gap. The magnitude of the maximum rate of
strain rises as the thickness of the liquid layer falls and the
recirculation region appears. In the recirculation region and
140
Newtonian Fluid
Separation angle θ degrees
(1)
120
Ca=0.11
Ca=0.3
Ca=0.6
Ca=1.0
Ca=1.2
Ca=1,5
100
80
(2)
60
40
20
10
(3)
0
H0 / tmin
2
3
4
5
6
7
8
9
Gap / Film Thickness, H0 / t
Fig. 18. Computed static contact angle as a function of capillary number and gap-to-thickness ratio. Newtonian liquid.
152
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
10
V
This work
Capillary Number, Ca =
0
Carvalho and Kheshgi (2000)
1
Viscocapillary Model (1980)
0.1
0.01
0.001
0
5
10
15
20
25
Gap / Film Thickness, H0 / t
Fig. 19. Critical gap-to-thickness ratio at the onset of the low-flow limit as a function of capillary number. Predictions are for Newtonian liquid.
in the coated liquid layer, the rate of deformation is small.
The maximum rate of deformation occurs close to the web
in the region underneath the downstream meniscus, where
the liquid is accelerated along its streamlines, and it is on
the order of 10–20 times the web speed-to-gap ratio. That
ratio is the characteristic deformation rate usually used in
previous analysis of slot coating flows. The type of deformation suffered by a liquid particle can be characterized
Fig. 20. Evolution of the deformation rate, in units of H0 /(1000V) as the film thickness falls. Ca = 0.3.
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
153
Fig. 21. Flow classification index near the downstream meniscus. Ca = 0.3.
with the flow classification index R, proposed by Astarita
[1]. Fig. 21 shows the R field near the downstream free
surface at Ca = 0.3. The vectors shown in the plot represent the direction of the highest positive linear deformation
(eigenvector associated with the positive eigenvalue of the
rate-of-strain tensor multiplied by the positive eigenvalue).
In the coated liquid layer, where the flow approaches a plug
flow, the deformation rate vanishes and R is high; in solid
body motion it would be unbounded. Under the die land,
R ≈ 1 and the eigenvector associated with the positive
eigenvalue of the rate-of-strain tensor are 45◦ from the flow
direction, indicating a shear-dominated flow. At the center
of the recirculation, the liquid motion approaches a solid
body rotation, and again R is high. The flow near the free
surface is dominated by extensional deformation, however;
the eigenvector associated with the positive eigenvalue of the
rate-of-strain tensor is aligned with the flow and R is small;
in pure extensional flow it would vanish. Although they have
analyzed situations at much higher flow rates (film thickness to gap ratio), Pasquali and Scriven [23] showed similar
behavior by defining a molecular shear and extensional
rates.
The high rate of strain and the extension-dominated deformation can lead to significantly higher stresses under the
meniscus in flows of extensional thickening liquids [27,10].
This is the central issue in the next section.
3.4.2. Effect of rheological properties
To examine the consequences of extensional thickening,
the shear viscosity was taken as constant with shear rate, as
in a Boger liquid:
η s = η0 .
In dilute polymer solutions the viscosity tends to plateau at
high enough extension rate that all of the coils are stretched
out (see [22]). Therefore, the dependence of the extensional
viscosity on the deformation rate is well described by a
Carreau-type relation, as proposed by Fuller et al. [11]:
ηu = η0 − (ηu∞ − η0 )[1 + (λu γ̇)2 ](nu −1)/2 ,
γ̇ ≡ tr(D2 ) is the second invariant of the rate-of-strain ten¯
sor D. The¯ parameters in this equation are η0 , the low-rate
¯¯
viscosity
in both shear and extension; ηu∞ , the limiting
high extension rate viscosity; λu , the characteristic time that
marks the onset of extensional thickening behavior; and nu ,
the exponent that describes the steepness of the variation of
the viscosity with extension rate.
The dimensionless parameters relevant to the non-Newtonian behavior of this extensionally thickening liquid are:
• Carreau number: W ≡ λu Vw /H0 ;
• extension-thickening exponent: nu ;
• ratio of extensional viscosities at high and low deformation rates: T ≡ ηu∞ /η0 .
The computed static contact angle as a function of the
gap-to-thickness ratio H0 /t and Carreau number W at
Ca = 0.3 and T = 10/3 is shown in Fig. 22. The effect
of the high-to-low deformation viscosity ratio T is portrayed in Fig. 23. When the coated film thickness is large
(low gap-to-thickness ratio), the rate of deformation is low
and the effect of extensional thickening is weak. As the
gap-to-thickness ratio rises, the deformation rates γ̇ become
larger, as shown in Fig. 20, and the effects of extensional
thickening become stronger. At a given capillary number
and flow rate, the computed static contact angle falls as
the extensional thickening behavior of the liquid becomes
stronger, i.e. higher W and T . Similar behavior has been reported by Lee et al. [18] when using differential constitutive
models. As a consequence of the change in the free surface
profile, the critical gap-to-thickness ratio at the onset of
the low-flow limit falls. The minimum film thickness that
can be coated rises as the liquid becomes more extensional
thickening. Fig. 24 shows the streamlines and free surface
154
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
Separation Angle θ degrees
140
Ca=0.3; Newtonian Model
120
Ca=0.3; T=10/3; W=1
Ca=0.3; T=10/3; W=10
100
80
60
40
W
20
10
0
H0 / tmin
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Gap / Film Thickness, H0 / t
Fig. 22. Computed static contact angle as a function of the gap-to-thickness ratio and pseudo-Weissenberg number. Ca = 0.3 and T = 10/3.
140
Separation Angle θ, degrees
Ca=0.3; Newtonian Model
120
Ca=0.3; W=1; T=5/3
Ca=0.3; W=1; T=10/3
100
Ca=0.3; W=1; T=50/3
80
60
40
T
20
10
0
H0 / tmin
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Gap / Film Thickness, H0 / t
Fig. 23. Computed static contact angle as a function of the gap-to-thickness ratio and the ratio of extensional viscosities at high and low deformation
rates T . Ca = 0.3 and W = 1.
Fig. 24. Txx component of the stress tensor, in Pascal, close to the free surface. Ca = 0.3 and H0 /t = 4.78.
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
155
Capillary Number, Ca = η0V/σ
10
Newtonian Model
W=1; T=5/3
W=10; T=5/3
W=100; T=5/3
W=1; T=10/3
1
W=10; T=10/3
W=1; T=50/3
0,1
3
4
5
6
7
8
9
Gap / Film Thickness, H0 / t
Fig. 25. Predicted critical conditions at the onset of low-flow limit for Newtonian and extensional thickening liquids.
configuration at Ca = 0.3 and H0 /t = 4.78, for both Newtonian and extensional thickening behavior (W = 1 and
T = 16.7). When the extensional viscosity of the liquid
rises with extension rate, the meniscus has to curve more
in order to balance the higher stresses that appear near the
free surface. Consequently, the static contact angle at the
downstream die lip falls.
The mechanism responsible for the change in the static
contact angle can be determined by analyzing the stress field
near the free surface. The stresses rise with Carreau number
due to the extensional behavior of the liquid. Fig. 24 also
shows the component Txx of the stress tensor in the region
close to the free surface. The rise in the extensional viscosity
near the free surface makes the stress in that region larger
than in the Newtonian case; a stress boundary layer is formed
close to the free surface, as predicted by Pasquali and Scriven
[23] and Lee et al. [18] using differential constitutive models.
This similarity in the stress field reinforces the statement
of Schunk and Scriven [27] that in the less concentrated
polymer solutions encountered in many coating flows, the
responses of shear thinning and extension thickening appear
to have more consequences than viscoelastic behavior.
The change in the force balance at the free surface due to
the extensional thickening of the liquid causes the solution
paths at W = 0 and T = 1 to cross the horizontal line of
θ = 10◦ in Figs. 22 and 23 at a lower gap-to-thickness ratio
than in the Newtonian flow. The onset of the low-flow limit
occurs at smaller gap-to-thickness ratio, i.e. thicker coated
layer. The critical conditions at the onset of the low-flow
limit are mapped in Fig. 25. The onset of the low-flow limit
when extensional thickening liquids are used occurs at larger
film thickness, as observed in the experiments presented in
the previous section.
4. Summary
Slot coating is one of the preferred methods for high
precision coating. There is an important operating limit,
known as the low-flow limit, when thin films are coated at
relatively high speeds. It is caused by the receding action of
the downstream free surface as the flow rate is reduced at
a fixed substrate speed. The operating parameters at which
the coating bead breaks have been determined by previous
researchers in the case of Newtonian liquids. However, most
of the liquids coated industrially are polymeric solutions
and dispersions, which are non-Newtonian. In general, polymer solutions may be shear-thinning, extension-thickening,
and develop viscoelastic stresses when sheared or extended.
These liquids can behave dramatically differently from
Newtonian liquids in coating flows. In the particular case of
very low concentrated polymer solutions of high molecular
weight (less than 0.1 wt.%), the response of extension thickening appear to have more consequences than viscoelastic
behavior per se (stress relaxation) (see [10,13,27]).
In this work the low-flow limit of polymeric solutions
was examined by both experiments and theory. A benchtop
slot coating apparatus was used to visualize the coating
bead at different operating conditions and observe the
breakup mechanism. Aqueous solutions of low molecular
weight PEG and high molecular weight PEO were used
in order to evaluate the effects of extensional-thickening
behavior. The flow visualization through the glass wall of
the rotating cylinder used as substrate revealed that close
to the onset of the low-flow limit, the downstream static
contact line recedes toward the feed slot at certain locations
across the width of the coating bead and the flow becomes
three-dimensional. Later, the air finger formed at the downstream lip penetrates all the way through the coating bead
to the vacuum chamber and one or more dry stripes appear.
The experiments also showed that, at a fixed capillary number, as the concentration of PEO rises and consequently the
liquid becomes more extensional-thickening, the minimum
coated thickness possible increases. The operating window
of the process is smaller when liquids with high extensional
viscosity are used.
The theoretical analysis consisted of solving the momentum and continuity equation system for steady,
156
O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156
two-dimensional, free surface flow including a generalized Newtonian model to describe the liquid rheological
behavior. The constitutive equation used relates the stress
to the rate-of-strain and relative-rate-of-rotation tensors.
This class of models is perhaps the simplest one that may
capture the different ways that polymer molecules behave
in extension-dominated and shear-dominated flow regions,
although it is not able to describe viscoelastic stresses per
se. The free boundary problem was solved by mapping the
unknown physical domain to a fixed reference domain. The
full set of differential equations, including the transformation mapping from the physical to the reference domain, was
solved all together by Galerkin/finite element method. The
theoretical low-flow limit was defined by the parameters at
which the static contact angle between the downstream free
surface and the die lip became less than 10◦ . In the case
of Newtonian liquids, the quantitative agreement between
the measured and the predicted parameters at the onset of
the low-flow limit is excellent. The results obtained with
the simple algebraic model revealed the same flow structure and stress field reported by previous work using more
sophisticated and computationally expensive differential
constitutive models. Moreover, the theoretical predictions
of low-flow limit of extensional-thickening liquids agree
qualitatively with the experimental observations of low
concentration polymer solutions, which presented high extensional viscosity and no measurable viscoelastic stresses.
Acknowledgements
The authors would like to thank Prof. Matteo Pasquali
for discussions on viscoelastic flows and for the suggestions
on the rheological characterization of the low concentration
PEO solutions. This work was funded by CNPq (Brazilian
Research Council, Grants 300242/98-0 and 467662/00-2)
and by the Industrial Partnership for Research in Interfacial
and Materials Engineering (IPRIME) of the University of
Minnesota.
References
[1] G. Astarita, Objective and generally applicable criteria for flow
classification, J. Non-Newtonian Fluid Mech. 6 (1979) 69–76.
[2] A.L. Beguin, Method of Coating Strip Material, US Patent no.
2,681,294 (1954).
[3] J.H. Bolstad, H.B. Keller, A multigrid continuation method for elliptic
problems with folds, SIAM J. Sci. Stat. Comput. 7 (1985) 1081–
1104.
[4] M.S. Carvalho, H.S. Khesghi, Low-flow limit in slot coating: theory
and experiments, AIChE J. 46 (2000) 1907–1917.
[5] Y. Christanti, L.M. Walker, Surface tension driven jet break up of
strain-hardening polymer solutions, J. Non-Newtonian Fluid Mech.
100 (2001) 9–26.
[6] J.J. Cooper-White, J.E. Fagan, V. Tirtaatmadja, D.R. Lester, D.V.
Boger, Drop formation dynamics of constant low-viscosity, elastic
fluids, J. Non-Newtonian Fluid Mech. 106 (2002) 29–59.
[7] J.M. De Santos, Two-phase cocurrent downflow through constricted
passages, PhD Thesis, University of Minnesota, MN, 1991.
[8] P. Dontula, M. Pasquali, L.E. Scriven, C.W. Macosko, Can extensional viscosity by measured in opposed-nozzle devices, Rheol. Acta
36 (1997) 429–448.
[9] P. Dontula, C.W. Macosko, L.E. Scriven, Model Elastic Liquids with
Water Soluble Polymer, AIChE J. 44 (1998) 1247–1255.
[10] P. Dontula, Polymer Solutions in Coating Flows, PhD Thesis, University of Minnesota, MN, 1999.
[11] G.G. Fuller, C.A. Cathey, B. Hubbard, B.E. Zebrowski, Extensional
viscosity measurements for low-viscosity fluids, J. Rheol. 31 (1987)
235–249.
[12] I.D. Gates, L.E. Scriven, Stability Analysis of Slot Coating Flows,
in: Presented at the AIChE 1996 Spring National Meeting, New
Orleans, 1996.
[13] J.E. Glass, Dynamics of roll spatter and tracking: part III. Importance
of extensional viscosities, J. Coat. Tech. 50 (641) (1978) 56–71.
[14] E.B. Gutoff, C.E. Kendrick, low-flow limit of coatability on a slide
coater, AIChE J. 28 (1987) 459–466.
[15] B.G. Higgins, L.E. Scriven, Capillary pressure and viscous pressure
drop set bounds on coating bead operatibility, Chem. Eng. Sci. 35
(1980) 673–682.
[16] H.S. Kheshgi, Profile equation for film flows at moderate Reynolds
number, AIChe J. 35 (10) (1989) 1719–1727.
[17] L. Landau, B. Levich, Dragging of a liquid by a moving plate, Acta
Physicochim. 17 (1942) 42.
[18] A.G. Lee, E.S.G. Shaqfeh, B. Khomami, A study of viscoelastic free
surface flows by the Finite Element Method: Hele-Shaw and Slot
Coating Flows, J. Non-Newtonian Fluid Mech. 108 (2002) 327–362.
[19] K.Y. Lee, T.J. Liu, Minimum web thickness in extrusion slot coating,
Chem. Eng. Sci. 47 (1992) 1703–1713.
[20] C.W. Macosko, Rheology, VCH, New York, 1994.
[21] C.Y. Ning, C.C. Tsai, T.J. Liu, The effect of polymer additives on
extrusion slot coating, Chem. Eng. Sci. 51 (12) (1996) 3289–3297.
[22] J.A. Odell, A. Keller, M.J. Miles, A method for studying flow-induced
polymer degradation: verification and chain halving, Poly. Comm.
24 (1983) 7–11.
[23] M. Pasquali, L.E. Scriven, Free surface flows of polymer solution
with models based on the conformation tensor, J. Non-Newtonian
Fluid Mech. 108 (2002) 363–409.
[24] K.J. Ruschak, Limiting flow in a pre-metered coating device, Chem.
Eng. Sci. 31 (1976) 1057–1060.
[25] L. Sartor, Slot coating: fluid mechanics and die design, PhD Thesis,
University of Minnesota, 1990.
[26] H. Saito, L.E. Scriven, Study of coating flow by finite element
method, J. Comput. Phys. 42 (1981) 53–76.
[27] P.R. Schunk, L.E. Scriven, Constitutive equation for modeling mixed
extension and shear in polymer solution processing, J. Rheol. 34–37
(1990) 1085–1119.
[28] W.J. Silliman, Viscous film flows with contact lines: finite element
simultions, a basis for stability assessment and design optimzation,
PhD Thesis, University of Minnesota, MN, 1979.
[29] R.L. Thompson, P.R. Souza Mendes, M.F. Naccache, A new constitutive equation and its performance in contraction flows, J.
Non-Newtonian Fluid Mech. 86 (1999) 375–388.
Скачать