Rheometry of paints with regard to roll coating process Olivier Cohu and Albert Magnin Citation: Journal of Rheology (1978-present) 39, 767 (1995); doi: 10.1122/1.550656 View online: http://dx.doi.org/10.1122/1.550656 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/39/4?ver=pdfcov Published by the The Society of Rheology Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 Rheometry of paints with regard to roll coating process Olivier Cohu and Albert Magnin'" Laboratoire de RMologie, B.P. 53, 38041 Grenoble Cedex 9 (France) URA CNRS 1510-Universite Joseph Fourier Grenoble I-Institut National Poly technique de Grenoble (Received 17 November 1994; accepted 19 April 1995) Synopsis The rheology of industrial paints is investigated with regard to roll coating (coil-coating) process. Within the nip between rotating rolls, the paint experiences high deformation rates over a very short time. Just after that, the liquid film has to level the defects generated between the rolls, and this occurs at low shear rates. Transientand steady-state rheological measurements are performed using shear and extensional rheometers with regard to the time and strain scales of the process. Paints are nearly Newtonian at high shear rates but frequently behave as thixotropic yield-stress fluids. The duration of the shearing between the rolls is found to be long enough to provide a total structural breakdown. Therefore, the steady-state viscosity is relevant for the determination of the film thickness which is depositedonto the web. Conversely, the structure recovery under low shear rates determines the rate of leveling and hence the surface quality of the finish. It is shown that the rate of recovery can vary considerably from a paint to one another. The thixotropy of the paints also affects the extensional measurements, but the results indicate the probable absence of actual extensional properties, even at high extension rates. © 1995 Society of Rheology. I. INTRODUCTION In most coating operations, one attempts to control both the thickness and the uniformity of the coated film. This can be achieved in a vast number of ways, such as dip coating, blade coating, or roll coating. The latter is the most common process used in the coil-coating industry, when a paint is to be deposited onto a metallic substrate such as aluminum or steel. An accurate control of the process requires the knowledge of the rheological properties of the fluid involved. Paints for coil-coating often exhibit a nonNewtonian behavior so that a specific rheometry able to mimic the time and strain scales of the process is needed. A typical coil-coater is depicted in Fig. 1. The relatively thick film formed on the pick-up roll divides between the pick-up roll and the applicator roll, which are counterrotating. The paint is then wiped from the applicator roll by the web, which moves in the opposite direction. Once deposited onto the web, the coating finally dries in the oven. Since the rubber-covered applicator roll is pressed against the web, only a trace of paint passes through the nip region, and the amount remaining on the applicator roll is negligible. Therefore, the amount of paint which is deposited onto the web is metered in the narrow gap between the applicator roll and the pick-up roll. The flow rate between these two counter-rotating rolls is controlled by an external load which presses the two a)Corresponding author. 1995toby The Society of Rheology, Inc. Redistribution© subject SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 J. Rheol, 39(4), July/August 1995 0148-6055195139(4 )n671191$7.00 767 11:09:01 COHU AND MAGNIN 768 MEDIUMRESIDENCE TIME LOW SHEAR RATE HIGH SHEAR RATE SHORT RESIDENCETIME HIGH EXTENSION RATE I Levelling of the surface defects I I Determination of the coaling thickness casteel web Oven f LONG RESIDENCE TIME LOW SHEA R RATE OR REST FIG. 1. Roll coater. cylinders against each other. Numerous studies attempted to predict the final film thickness which is deposited onto the web. The metering flow of Newtonian fluids between two counter-rotating cylinders was investigated, e.g., by Pitts and Greiller (1961) and Savage (1982). The most extensive analysis was presented by Coyle (1984), who also considered shear-thinning fluids (see also Coyle et al., 1986, 1987). All these studies focused on the simple case of two rigid rolls and very few papers dealing with a deformable rubber roll have been published. Coyle (1988) carried out both theoretical and experimental studies. Experimental data were also reported by Kang et at. (1991), who considered both Newtonian and non-Newtonian fluids. When a fluid splits between two rolls moving in the same direction at the nip, the films on the rolls often exhibit a ribbed pattern due to flow instability (see, e.g., Pitts and Greiller, 1961; Coyle et at. 1990). As was noticed by Mill and South (1967) or Matsuda and Brendley (1979), the ribbing instability is inevitable except at unacceptably low speeds. The ribbed pattern may be carried out to the finish, unless the coating levels sufficiently before drying in the oven. Therefore, the surface quality of the finish greatly depends on the ability of the paint to level rapidly. The leveling of a Newtonian fluid was investigated by Orchard (1962) and Khesghi and Scriven (1988). Since paints are often thixotropic, numerous studies have been carried out to connect the levelability of the paints and their rheological properties (Dodge, 1972; Matsuda and Brendley, 1979; Patton, 1978; Kristiansen, 199 I). These authors pointed out that the major difficulty is to provide pertinent rheological characterizations in order to simulate the leveling flow correctly. Any analysis of the roll-coating process presupposes that the rheological properties of the fluid with regard to the strains encountered in the whole process can be determined. The rheological design of paints for coil-coating must consider on one hand the high deformation rates and the short time scale of the flow between two rotating rolls and on Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 769 RHEOLOGY FOR ROLL COATING pressure distribution (Swift-Steiber exit boundary condition) FlG. 2. Schematic drawing of the lIow kinematics between two rigid rolls in the symmetric case. the other hand the low deformation rates and the longer time scale of the leveling flow. These time and strain scales are evaluated in the next section. II. DEFORMATION RATES AND TIME SCALES OF THE ROLL COATING PROCESS A. Scales of the flow between two counter-rotating rolls With two rigid rolls of radii R I and R z, separated by a gap 2H0 and moving respectively at speeds V I and V z in the same direction at the nip, one can define the mean velocity V and the equivalent radius R as (I) and R == 2R 1R z/(R I +Rz). (2) The flow between both rolls has two components. The Poiseuille contribution is generated by the pression field induced by the converging/diverging geometry. This leads to a parabolic velocity profile in the gap, with a maximum velocity of about 1.5 V (Coyle, 1984). This is shown in Fig. 2 in the symmetric case where VI =V2 =V and R I = R 2 = R, and here the shear rate in the gap can be approximated by YPoiseuille "'" V 2H . o Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 11:09:01 (3) 193.48.255.141 On: Fri, 03 Jan 2014 no COHU AND MAGNIN Conversely, when both rolls move with different peripherical speeds, a Couette contribution to the flow arises, which induces an additional shear rate in the gap V 1-V2 'YCouene.... 2H (4) o Finally, the characteristic shear rate of the flow between two counter-rotative rigid rolls can be approximated, in the general case, by V .y - 2H o ｾ V 1-V2 + Poiseuille contribution 2Ho (5) Couette contribution The time of transit of the fluid within the narrow gap between two cylinders is given roughly by the ratio of the width 8 of the lubricated contact to the mean velocity V. With two rigid cylinders and using the Swift-Steiber (Reynolds) exit boundary condition, the lubrication theory predicts the distance between the exit meniscus (denoted S in Fig. 2) and the center of the gap to be approximately 0.67 (RHo)ln (Coyle, 1984). The contact width 8 is at least double this, and then the time of transit t of the fluid in the shearing region between the rolls verifies (RHo) 112 t ｾ (6) V 1.5 Greener and Middleman (1975) and Glass (1978) showed that roll coating flows also have a significant extensional component. Soules et aJ. (1988) proposed an estimation of the extension rate in the film splitting region which was based on experimental observations. However, the maximum extension rate takes place within the nip region. At point S' in Fig. 2, the zero pressure gradient imposes the fluid velocity to equal V. Conversely, at the center of the nip, the fluid velocity is approximately 1.5 V. From lubrication theory (Coyle, 1984), the distance between these two points is about 0.67 (RHO) 112. Therefore, the extension rate between these two points is given by , E"" 0.5 V 0.67(RH ) 1I2 o .... 0 75 . V (RH ) 1I2 • o (7) For a typical coil-coating process, R, H 0, and V 1 and V 2 equal respectively 0.10 m, 30 ""m, 2 mis, and 0.5 mis, and then the strain and time scales of the flow between the rolls can be approximated in the case of rigid rolls from Eqs. (5)-(7): (rigid rolls). (8) When a deformable rubber roll is involved, the flow kinematics is more complex, and then the time and strain scales of the flow are more difficult to estimate. Since the two rolls are pressed against each other, their surface tends to become parallel and the peak. pressure decreases considerably (Pangalos et al. 1985). This causes the Poiseuille contribution to the shear rate to drop several orders of magnitude, and then the shear rate in the gap reduces to the Couette component Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 RHEOLOGY FOR ROLL COATING "I"'" 771 (9) where 2H o is here the nip spacing in the parallel region. Conversely, the development of a parallel gap significantly increases the contact width 8 which tends to the Hertzian contact value. The latter is an order of magnitude greater than the width of the lubricated contact between two rigid cylinders (Coyle, 1992). The result is a significant decrease of the extension rate and a significant increase of the time scale of the flow, see Eqs. (6) and (7). Thus, with the numerical data used earlier, the order of magnitude of the time and strain scales of the flow between the pick-up roll and the applicator roll for an actual coil-coating process should be estimated according to 4 -lj "I"'" 2Xl 0 s . i""'5Xl0 1 s- 1 (actual coil-coater involving a rubber roll). (10) t;;' 2XlO- 2 s B. Leveling flow The leveling of a ribbed film of paint takes place at low shear rates of 10-10- 2 s-l (Dodge, 1972, Matsuda and Brendley, 1979, Kristiansen, 1991). The time scale to be considered is about 20 s, which is the time necessary for the web to travel from the rolls to the oven. An important point is that the leveling flow takes place immediately after the intense shearing that occurs between the rolls. III. EXPERIMENT The rheology of paints designed for coil-coating applications was investigated with rotational and capillary rheometers. The Rheometries RFX rheometer, which was designed for both shear and extensional measurements, was also used. These instruments allowed us to reach the conditions encountered by the paints during the complete coilcoating process. The rheological properties of the paints were measured at shear rates between 10- 3 and 105 s -I and extension rates between 101 and 104 s -1. Specific test procedures were performed in order to investigate the transient flows which occur at the inlet and the outlet of a nip. A paint used in coil-coating processes is a dispersion of either mineral or organic pigments in an organic solvent with a polymeric binder (Bonnebat et al., 1994). Additives such as colloidal silica are sometimes added to prevent pigment settling. The polymer binder is either dispersed or dissolved. In the latter case, its molecular weight is low. Because of the wide variety of formulations used in the coil-coating industry, an exhaustive study is impossible, and therefore six representative industrial paints were tested. Three types of behavior were found which are illustrated by paints 1,2, and 3. The solid content of these materials was about 45% and the particles were smaller than 5 ,um. IV. STEADY-STATE SHEAR RHEOLOGY OF COIL-COATING PAINTS For the determination of the steady state viscosity functions of paints I, 2, and 3, a Carrimed Weissenberg (controlled strain) rheometer was first used. This instrument was equipped with a cone-and-plate cell for shear rates lower than 316 s-1 (cone angle: 10 , diameter: 50 or 75 rom). Solvent evaporation was avoided by saturating the atmosphere around the free surface of the sample. For shear rates between 316 and 7500 s -1, we used a Couette geometry (bob outer radius 15 mm, rotating cup inner radius 15.3 rom, height Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 772 COHU AND MAGN IN 10' o T s 20"C paint 1 • paint 2 A paint 3 - - model sq. (11) 10" 10. 3 10. 2 10. 1 10° 10' shear r.te 10 2 10' 10 4 105 (.1) FIG. 3. Steady-state viscosity as a function of shear rate for paints I, 2, and 3, at T = 20°C. 50 nun). The temperature was measured to an accuracy of :to.1 °C with a thermocouple inserted into the tools at the contact of the sample. In order to plot the viscosity functions at the standard temperature of 20°C, each point was corrected as described later (see Sec. IV A). A Gottfert 2001 capillary rheometer was used to measure viscosity functions at shear rates higher than 7500 s-l. Glass capillaries of diameter 0.41 nun and length over diameter ratio of 118 to 191 were used. Since this ratio was high, it was observed that the entrance pressure could be neglected. Conversely, shear heating could be important, so that adequate corrections were required. The power dissipated by shear effects was approximated from the apparent viscosity and the apparent shear rate. Then, the temperature of the fluid at the exit of the capillary was estimated from the initial temperature in the reservoir by assuming the die to be adiabatic. The average temperature between the entrance and the exit of the capillary was finally considered to correct the data, according to the law described in Sec. IV A. With this correction, the data were found to recover well with those obtained with rotational rheometers. The steady-state viscosity functions of paints 1, 2, and 3 at 20°C are plotted in Fig. 3 for shear rates between 10- 3 and 105 s- I. It can be seen that the different curves intersect, which shows that the rheology of paints must be investigated over the whole range of shear rates. With paint 1, shear rates below 0.1 s-I did not generate a measurable stress. Above 0.1 s - I , this paint is practically Newtonian, with a viscosity of about 0.65 Pa.s. Conversely, paints 2 and 3 are non-Newtonian and their viscosity follows a generalized Casson equation = 7f - n TO T Y- ( yn + n TJoo ) lin , (11) which introduces the yield stress TO and the high-shear-rate Newtonian viscosity TJoo. [Note that Eq. (11) reduces to that of a Newtonian fluid if TO = 0 and n = I.] The values obtained for 7b and TJoo with paints 2 and 3 are reported in Table I. It should be noted that paint 2 reaches its Newtonian plateau when y ｾ 103 s -I, while paint 3 is still slightly shear-thinning at .y = 105 s- 1. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 RHEOLOGY FOR ROLL COATING 773 TABLE I. Rheological parameters of the paints at T = 20°C. Paint No. Infinite shear rate viscosity 7}.0 (Pa.s) I 2 0.65 0.25 0.10 3 Yield stress ..o Viscosity function Temperature (Pa) power index n coefficient a T =0 =1 0.40 0.25 0.15 0.25 1.067 1.059 1.042 The shape of the viscosity function of paints 2 and 3 is probably due to the presence of additives which prevent pigment settling. Such additives are probably missing in paint 1 since significant pigment settling could be observed with this paint. A. Temperature dependence The influence of temperature on the steady-state rheology of paints was investigated using a controlled stress rheometer Carrimed CSL lOO, with a cone-and-plate geometry (cone angle: 10, diameter 60 rom). This rheometer was equipped with a Peltier system allowing an accurate control (:±:: 0.1 "C] of the sample temperature. Following previous experiments, care was taken to prevent solvent evaporation. In industry, paints are applied at a temperature between 15°C and 30 "C. In their investigation of the rheology of paper coatings, Laun and Hirsh (1989) found that a variation of the temperature did not affect the shape of the viscosity function. The whole curve was simply shifted parallel to a straight line of slope -1 in the log-log plot, with a shift factor A(T) such as a temperature invariant mastercurve could be obtained by plotting TjA(T) vs A(T)Y. We observed the same effect with coil-coating paints, and the shift factor was found to follow Eq. (12) (T-Tref) (12) A (T) "'" "t in the range of temperature 15 °C-30 "C. Here Tref is a reference temperature and or is a constant which depends only on the paint considered. This is illustrated in Fig. 4 for 10' ;:;: T ｲＮｾ ro' = 20°C "+ o ' l=' f 10° ..... • ". -'. .'b.... 0 lil ,1 _-- .... " '> :::I Ts15.0·C Ts22.5·C T=3O.0·C _. ••••••••• generalized Casson eqn. atT.... (dala Table 1) 10° 10' reduced snear rste : 10 2 10' Y.,. (T·Trof) FIG. 4. Temperature invariant mastercurve (Tref = 20°C) of the viscosity function of paint 2 (a T = 1.059). Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 COHU AND MAGNIN 774 shear rate pres hear rate Y Yp rest time - timet o FIG. 5. Step shear-rate test simulating the paint flow between rolls. paint 2. The temperature coefficient aT was found to be between 1.040 and 1.070 (see Table I), For a Newtonian paint such as paint I, with" independent from y, this result simply indicates that the temperature dependence of the viscosity is in the range of 4%-7% per degree Celsius. Equation (12) was used to for the temperature corrections in the rheological measurements as mentioned previously. V. TRANSIENT FLOW AT THE INLET OF A NIP BETWEEN TWO ROLLS A. Shear stress overshoots in step shear-rate tests At the inlet of a nip between two rolls, paints are subjected to a sudden change of shear rate within a very short time. We reproduced these conditions with the Carrimed Weissenberg rheometer equipped with the Couette geometry described above: after preshearing at I(}()() s - I, the paints were left at rest for 30 minutes, allowing the material to recover a great extent of its initial structure, as shown in Sec. VI. Then a shear rate of 7500 s -1 was suddenly imposed (Fig. 5). The rheometer was equipped with a piezoelectric transducer with a negligible response time. Conversely, the time necessary for the rheometer to reach the nominal share rate of 7500 s -1 was found to be approximately 40 ms with a Newtonian silicon oil of viscosity 0.3 Pa.s (Fig. 6). Further experiments at lower nominal shear rates were performed with a Rheometries RMS 800 controlled strain rheometer, equipped with a TRUOO transducer and a cone and plate geometry (cone angle 0.9°, diameter 50 mm). The response time of this system was found to be less than 30 ms for a step shear rate from zero to 316 s -1. When subjected to a sudden change of shear rate from 0 to 7500 s -I, the paints exhibited practically the same response as the Newtonian silicon oil (Fig. 6). No elastic behavior was discernible, and no thixotropic overshoot was observed, since the steadystate regime was reached before 40 ms. In order to observe the thixotropic overshoots on a time scale larger than the response time of the system, experiments were performed at lower shear rates. We could then relate the time necessary to' reach the steady-state regime to the total applied deformation. It can be seen in Fig. 7 that the stress level measured after a shear strain of 10 does not exceed the steady-state value by more than 30% for y between 10 and 100 s- 1. Moreover, for i' = 316 s -I, the stress level measured after a shear strain of 10 exceeds the steady-state value by only 10% or even less. Therefore, one can infer that at higher shear rates such as those encountered between two rolls, the steady-state regime would be reached practically after a total deformation of 10. Note that for y = 7500 s -1, a shear strain of 10 is reached 1.3 ms after start-up, which would explain the absence of thixotropic overshoots in Fig. 6. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 RHEOLOGY FOR ROLL COATING 775 1.2 I j ----L!<lo ｾＧ Ｂ ｲＡＭｾ Ｇ Ａ ｾ｢ ｾＮ o () (), ! 0 "100000 0 0 0 0 j 00 ! ----. ﾷ｟ＭＡ ﾷ｟Ｍｾｴ __··,·__· ! til shear rate. 7500 s" 0.8 0.6 ｾＭ -e--L.-----+----- ...! i 4t! I aller preshear1000 .. and 1800 8 of rest , r-r:..m-r-t=I:::=- 0.4 0.2 o silicon oil (0.3 Pas) ! o o 40 80 120 200 180 Urne (1M) FIG. 6. Reduced stress vs time following a sudden change of shear rate from zero to 7500 s-I at T = 20°C :!0.5 0. Comparison between paint 2 and a Newtonian silicon oil. The first 40 ms correspond to the rheometer acceleration and should not be taken into account. From Eq. (8), it can be seen that a fluid flowing between two rigid rolls is subjected to total deformation " of (13) Equation (10) similarly reveals that the total shear strain is higher when a deformable roll is involved. Hence, despite the numerous assumptions that have been made, one can expect that the steady-state regime is practically reached at the outlet of the nip. Note that from Eqs. (5) and (6), the total shear strain (.y r) does not depend on the average roll velocity V, so that this result would be valid at any speed. 2 ; - s h e a r rate .10 s" ---v-shear I1Il8 .100 s" 1.5 !I---""".r-.-----+'--- - ---- shear rate • 316 s" I ｾＮ ｾ I I Ｎ｟Ｎ｟Ｎ｟ｾ｟Ｎ｟Ｍ｟ I ......•- ... Ioc i I :a .---i-----t----r-"--... --.....---j I· I• ,, , ---_ , ., 0.5 I ..... , o ,. " ,. " o , .. i I : ｾＢ｡ＧＳＱＶ -----t--..--aIt.r ... i I s., _ and 1800 s of rest I i i 5 10 15 20 25 30 ..... str8ln FlG. 7. Reduced stress vs time following a sudden change of shear rate from zero 10 various shear rates at T = 25°C :!:0.1 0 for paint 2. The ascending part of the curves correspond to !he rheometer acceleration and should not be taken into account. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 COHU AND MAGNIN 776 Torque transducer \ -, Syringe pump FIG. 8. Schematic drawing of the RFX rheometer in shear configuration. B. Rheometry of paints in a short transient high-shear flow The high-shear flow of short time scale that occurs between two rolls was simulated using the high shear fixture of the Rheometries RFX rheometer. The principle of this instrument is similar to that of the high-shear/short-time "HSM" rheometer described by Wallpott et al. (1991). As shown in Fig. 8, it consists of a hollow cylinder (tube), of inner diameter D 1 , through which the fluid is pumped, and a smaller coaxial cylinder (pin), of diameter D2, on which the force is measured. With the axes properly aligned, the pin and tube from an annular channel whose length L can be varied according to the depth of insertion of the pin. Both cylinders are immersed in a thermostatically controlled beaker of fluid. The syringe barrel is withdrawn at a controlled rate, aspirating the fluid through the annular channel at the desired throughput Q, while a TRT 200 torque transducer measures the force F on the pin arm. Since the gap h "'" (D j - D 2)12 is small compared to the radii, the flow is equivalent to the flow through a rectangular slit of length L and width 7T{D j + D2)/2 (Krieger and Huang, 1994) and then the shear rate and the shear stress at the slit wall are given respectively by Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 RHEOLOGY FOR ROLL COATING 777 (14) and F 7" = ---,---, ｾ D ZL7T( (15) +1 ) ' In this configuration, the mean time during which an elementary volume of fluid is sheared can be estimated from the flow rate Q, and can be written as t = 6(IJh)y -[, (16) using Eq. (14). Experiments were performed with a specially designed fixture of diameters D 1 = 1.100 mm and D z = 0.726 mm. The pin and tube axis alignment was checked using the fact that perfect alignment corresponds to maximum force on the transducer (Krieger and Mode, 1992). With the small gap of 187 flm the high shear rate of 104 s -1 could be reached. The nominal length of the fixture was chosen to be 2 mm, so that the duration of the shearing of an elementary volume of fluid could be less than 10 ms [Eq. (16)]. With this instrument, the rheological behavior of the paints could therefore be directly investigated in time and strain scales of the flow between rotating rolls. The reliability of the instrument was first tested with Newtonian glycerin-water solutions of known viscosity. From Krieger and Huang (1994), it was checked that no kinetic energy corrections were needed in our experiments. Inversely, end effects of the Bagley type had to be corrected. The entrance force was determined by extrapolating to L1h = 0 the straight line of F vs L1h. Once the data corrected, the accuracy of the measurements was found to be better than 5% for y comprise between 103 and 104 s-[ in the range of viscosities 0.2-1 Pa.s. Paints were tested after a rest period of 30 minutes, allowing a large structure recovery as mentioned previously. Though low LI h ratios were used, the Bagley plots F vs LI h were found to be straight lines for y comprise between 103 and 104 s -1, even in the case of the thixotropic paints 2 and 3 (Fig. 9). Ends correction could therefore be easily determined from the intersection of these lines with the ordinate axis. The results are shown in Fig. 10. Even in the case of the thixotropic paints 2 and 3, the data obtained in the RFX brief shear flow were found to coincide with the steady-state viscosity curves. This indicates that the networks likely to develop within these paints are broken before or just at the entrance of the controlled shearing region. Such a result confirms the idea that such networks are very weak. In other words, the time necessary to destroy these networks is far shorter than the time scale of the rheometric flow. This confirms that the steady state is relevant to describe the flow properties of the paint during the brief but intense shearing that occurs between two rolls. VI. STRESS RECOVERY UNDER LOW SHEAR RATES Just after the intense shearing which occurs within the nip, the paint levels at low shear rates. Thixotropic structure recovery under low shear rates, following intense preshearing, was investigated with step shear-rate experiments (Fig. 11). These experiments were performed with the Carrimed Weissenberg rheometer, equipped with a torsion bar transducer allowing a torque resolution of 1 flN.m. This transducer had to be protected against overloads during the preshear phase. The torsion bar transducer has a long reRedistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 COHU AND MAGNIN 778 3.5 _10001-1 3 _31621-1 _10000 ..1 2.5 § .. ! s 2 1.5 0.5 0 2 0 • 4 Llh • 10 12 14 nG. 9. "Bagley plots" of paint 3 at several shear rates (RFX rheometer in shear configuration). T = 20·C :to.5°. sponse time of about 2 s, but the time scale to be considered here is far longer since the duration of the leveling phase is about 20 s. A high-shear cone-and-plate cell, similar to the apparatus described by Laun and Hirsh (1989), was used (cone angle 1°, diameter 50 mm). This system was designed to confine the paint between the cone and the plate even at high rotation speeds. In this configuration, the preshear time had to be minimized to avoid the particle segregation due to centrifugal forces. Paint I is not considered here since shear rates below 0.1 s - I did not generate a measurable stress with this paint, as mentioned earlier. With paints 2 and 3, the stress level observed at 0.1 s -1 immediately after preshearing is at the limit of the system resolution. However, the thixotropic stress recovery is clearly visible, as shown in Figs. 10° • ii' 0 l ｾ i rotational rheornel8rs (steady state) RFX (high lhearfixlure) • '5 1 ｾ . ｾＮ i I ｾ • i> • 0 • 10" 10· 10 ' 10 4 ...... ,... ("1) nG. 10. Viscosity function of paint 3 at 20·C :to.5°: comparison between the results obtained with the RFX rheometer and the steady-state viscosity measured with rotational rheometers. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 RHEOLOGY FOR ROLL COATING Ｎｾ preshear rate 779 Y D Ql "! ti .!III "C ! shear ra1e )' llmet o FIG. 11. Step shear-rate test simulating the paint history after its ftow between the rolls. 12 and 13. It was found that a relatively low preshear (100 s-lor even 10 s-1) was sufficient to provide a complete structural breakdown since increasing the preshear rate from 100 to 7500 s - I did not change the shape of the recovery curve anymore. This confirms that the networks likely to develop within the paints are very weak, which explains the rapid decay of any transient flows at high shear rates, as was observed previously. With both paints, the time necessary to reach the steady-state regime is longer than the time scale of the leveling phase, which means that thixotropy plays a major role in the leveling flow. Hence it is clear that the determination of the steady-state viscosity is irrelevant to describe the rheological behavior of the paints during the leveling phase of the coating process. Another important result arising from Figs. 12 and 13 resides in the comparison between the rate of recovery of the two paints: The time of structure recovery at 10- 1 s-1 1 .. to- 0.8 X :! 1; <> • 0.6 I c•0 'ii c prnhear. 10.1 prnhear- 100 preshear. 1000 s·l preshear. 7500s-t .-1 1 ,..-----------, J 0.4 ZOOM I is 0.2 20 o o 50 100 150 200 250 300 time (s) FIG. 12. Reduced stress vs time following a sudden change of shear rate from various shear rates to 10-] s(paint 2. T = 20.5 ·C±O.IO). I Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 780 COHU AND MAGNIN 1 • ｾ :I • 0.8 ! •:" 0.6 'E 0 Ii c E • :a ••.••_.j.. 0.4 ------- ＭｾＧＭ ｪ 0.2 . x I 0 preshear _ 100 s-t preshear = 1000 s-l preshear • 7500 5-1 j • _ ¢> <:tl 0 0 8 4 II.... 12 16 20 (s) FIG. 13. Reduced stress vs time following a sudden change of shear rate from various shear rates to 10- 1 S-I (paint 3, T = 20.5°C±0.lO). is ten times longer for paint 2 than for paint 3. Similar results were obtained at lower shear rates 10- 2_10- 3 s-I (not shown). From an industrial point of view, paints 2 and 3 differ since the viscosity of paint 2 remains low during the whole leveling phase. Therefore, the ribbing defects generated between the rolls are expected to disappear much faster with paint 2 than with paint 3. VII. EXTENSIONAL RHEOMETRY OF COIL-COATING PAINTS The extensional properties of paints were investigated using the Rheometries RFX extensional rheometer, which is based on the opposed jet device (Fuller et at. 1987). Figure 14 shows a schematic drawing of the instrument. The fluid is either sucked or expulsed at a controlled flow rate Q through two immersed nozzles of diameter D separated by a gap G. One of the nozzles is connected to a force rebalance transducer. The equations for the apparent strain rate € and the apparent normal stress (TIl - T22) have been derived by Fuller et at. (1987) and can be written as . 8Q E - ------.". - «ctr ' 4F TIl - T22 = ----;::z , rrD (17) (18) where F is the force exerted on the nozzles. Experiments were conducted with three pairs of nozzles, of diameter 2, 1, and 0.5 mm. Following Schunk et at. (1990), the gap between the nozzles was set equal to their diameter, (i.e., G = D) in order to minimize the shear contribution to the flow. In order to check the validity of the technique, experiments were first conducted with Newtonian, glycerin-water solutions of viscosity comprise between 0.3 and 1.1 Pa.s. Theoretically, the extensional viscosity should be three times the shear viscosity for Newtonian materials. This is called the Trouton ratio and it is used as a reference for extensional viscosity measurements. Experimentally, very scattered data were obtained Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 RHEOLOGY FOR ROLL COATING 781 Torque transducer Syringe pumps Extensional flow FIG. 14. Schematic drawing of the RFX rheometer in extensional configuration. with Newtonian solutions of viscosity close to that of the paints (Fig. 15) even though better results were obtained with more viscous materials. Nevertheless, the mean Trouton ratio was always found to be about 4.5, which is much higher than the theoretical value of 3. Trouton ratios between 4 and 5 were also obtained by Mode (1992) with Newtonian mineral oils, and this discrepancy is probably due to the shear contribution to the flow (Schunk et at. 1990). A significant discrepancy between the results obtained in compressional and extensional tests (denoted respectively efflux and influx in Figs. 15-18) was also found. The apparent Trouton ratios obtained in compressional tests. were usually lower than those obtained in extensional tests. No satisfactory explanation was found about this. Since scattered and shifted data were obtained with Newtonian fluids of shear viscosity close to that of the paints (Fig. 15), the extensional properties of the coatings could only be investigated qualitatively. The wide Trouton ratio window of 3-7 which was obtained with Newtonian fluids was therefore considered as a reference. With paint 1, the experimental values of the Trouton ratio are located in the reference window of 3- 7, which means that this paint behaves like a Newtonian fluid even in an extensional flow (Fig. 16). The results are quite different with the thixotropic paints 2 and 3. With these paints, the curves obtained with different sets of nozzles do not intersect, as shown in Fig. 17. This was not observed with Newtonian fluids. Each of these curves Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 782 COHU AND MAGNIN Glycerin I water c o o :; ｾ - - theoretical _ nozzles .. 2mm (infkJx) - - e.- • nozzles .. 1 mm(e1fux) . . - nozzleS" O.5mm (infkJx) - - . - - nouleu 2mm (e1fux) - _ _ - - nozzleS" O.5mm (eIllux) - nozzles .. 1mm (innux) 10 2 10 3 extension rate (S-1) FIG. 15. Apparent Trouton ratio as a function of apparent extension rate for a Newtonian glycerin/water solution of shear viscosity 0.31 Pa.s at T = 20°C :'::0.5°. shows an apparent decrease ofthe Trouton ratio from about 20-30 to the upper limit of the reference window. Moreover, the apparent Trouton ratio was found to depend only on the total flow rate Q between the nozzles, as shown in Fig. 18. This is quite surprising since one single value of Q corresponded to several values of the extension rate E, by changing the diameter and the gap of the nozzles [see Eq. (17)]. The same effect was recently observed by Anklam et al. (1994) with water-in-oil emulsions exhibiting a yield stress. They showed that forces on the nozzle arm arising from the yield stress in the fluid dominate the measurements. They concluded that the measurements of elongational viscosities of fluids with significant yield stress are not possible with the present apparatus. These remarks hold with paints 2 and 3 which are yield-stress fluids. However, one may say that when the flow rate increases, the forces ｾＭ Ｚ Ｍ ｾ c o :; o , Paint 1 (N Newtonian) ... ---0--- ｾ _ nozzles .. 2mm (inllux} _ -nozzles" lmm (influx) - . . - ooules .. O.5mm (influx) - -0 - - nozzles .. 2mm (efflux) - -0-· nozzles" lmm (efflux) _ - • nozzles II O.5mm (efflux) extension rale (S-l) FIG. 16. Apparent Trouton ratio as a function of apparent extension rate for paint t at T = 20 DC :'::0.5°. The horizontal solid lines indicate the reference window which encloses the experimental data for Newtonian fluids. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 RHEOLOGY FOR ROLL COATING 783 Paint 2 (thixotropic) -·0··0·-0 nozzles" 2mm (infux) ·-0-- nozzles II 2mm (eftlux) ...... - nozzles" 1mm (infux) • ·0 •• nozzles II lmm (eftlux) _ - --e,- - nozzles" 0.5mm (elllux) . . - nozzles" 0.5rnm (infux) 10 2 10 3 extension rBte (5-') FIG. 17. Apparent Trouton ratia as a function of apparent extension rate for paint 2 at T = 20°C ±0.5°. The horizontal solid lines indicate the reference window which encloses the experimental data for Newtonian fluids. arising for the local velocity field around the nozzles increase, and then the contribution from the external flow field forces arising from the yield stress decreases. In other words, extensional viscosity measurements under high flow rates should be less affected by the presence of a yield stress. At high flow rates, the apparent Trouton ratio obtained with paints 2 and 3 tends to the reference window which was obtained with Newtonian fluids. Therefore, one might expect that these paints do not exhibit a specific extensional behavior. Paint 2 (thixotropic) _ nozzles II 2mm (ef1lUX) ...... -nozzles II 1mm(ef1kJx) - . . . nozzles" O.5mm(ef1lux) o ! c o -; 10 ' -- e I- 10 2 10 3 flow rate Q (mm3/s) FIG. 18. Apparent Troutan ratio as a function of the flow rate between the nozzles for paint 2 at T = 20°C ±O.5°. The horizontal solid lines indicate the reference window which enclosed the experimental data for Newtonian fluids. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 784 COHU AND MAGNIN VIII. CONCLUSIONS With specific rheological measurements, the shear and extensional rheology of paints have been investigated in the time and deformation-rate scales of the coil-coating process. Three representative rheological behaviors were distinguished. Paints were found to be nearly Newtonian at high shear rates, but a low yield stress of about 0.2 Pa was observed in certain cases. This yield stress may be due to the presence of additives in the paints used to prevent pigment settling. These additives induce thixotropic behavior, so transient rheological measurements are absolutely necessary. The final film thickness of the coating is governed by the high-shear rheology of the paint over a very short time scale. The experiments that have been carried out show that the duration of the shear flow between the rolls is long enough to provide a complete structural breakdown within the paints. Furthermore, no elongational component was detected over the strain scale of this flow. Therefore, the high shear-rate steady-state viscosity is relevant to describe the flow properties of the paint during its passage between two rolls. Instabilities in the flow of a liquid between two counter-rotating rollers generates an undesirable ribbed pattern on the film and then the smoothness of the finish greatly depends on the ability of the paint to level rapidly. The leveling rate is governed by the low shear-rate viscosity of the paint following intense preshearing. With thix.otropic fluids, this transient viscosity can be far lower than the steady-state value and increases with time as the structure builds up. From a paint to one another, the rate of structure buildup can vary by a factor 10 or more. The determination of this parameter is thought to be a useful tool for the selection of industrial paints with regard to leveling criterion (Cohu and Magnin, 1995; Cohu, 1995). The rheological tests that have been proposed ultimately lead to the knowledge of all the fluid properties that determine the runnability of the complete coil-coating process. Additionally, it is believed that this methodology is not limited to the particular case of coil-coating paints but should be relevant in the study of any fluid applied by roll coating. ACKNOWLEDGMENTS The authors are indebted to Claude Bonnebat (SOLLAC-C.E.D., Montataire, France) who initiated this work and provided helpful advice. For the use of the RFX rheometer, we are grateful to RHEOMETRICS Inc., Piscataway, New Jersey, USA, who graciously lent us the instrument. In particular, we would like to acknowledge Paul Mode for technical support and fruitful discussions. The encouragements from Pro Jean-Michel Piau were very much appreciated. We also wish to thank Jeff Goshawk for correcting the English text. This work forms part of the Ph.D. project "Rheologie des peintures et precede de couchage au rouleau," which is being supported by SOLLAC and by the ANRT (Association Nationale de la Recherche Technique). References Anklam. M. R.. G. G. Warr, and R. K. Prud'homme, "The use of opposed nozzles configuration in the measurements of the extensional rheological properties of emulsions," 1. Rheo\. 38, 797-810 (1994). Bonnebat, C .. R. Hellouin, F. Carrara. and F. Chichyvan, Les revetements organiques, in Le livre de racier, edited by G. Beranger, G. Henry, and G. Sanz with Sollac, Lavoisier Tee and Doc (1994), pp. 713-756. Cohu, 0., Rheologic des peintures et precede de couehage au rouleau. Thesis. Institut National Polytechnique de Grenoble. France (1995). Cohu. o. and A. Magnin, "The levelling of thixotropic coatings." Prog. Org. Coat. (in press, 1995). Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01 785 RHEOLOGY FOR ROLL COATING Coyle, D. J., "The fluid mechanics of roll coating: Steady flows, stability and rheology," Ph.D. thesis, University of Minnesota, 1984. Coyle, D. 1., "Experimental studies of flow between deformable rolls," AIChE Spring National Meeting, New Orleans. LA (March, 1988). Coyle, D. J., "Roll coating," in Modem Coating and Drying Technology, edited by E. Cohen and E. Gutoff (VCH publishers, New York. 1992), pp. 63 -116. Coyle, D. J" C. W. Macosko, and L. E, Scriven, "Film-splitting flows in forward roll coating," J. Fluid Mech. 171, 183-207 (1986). Coyle, D. J.. C. W. Macosko, and L. E. Scriven, "Film-splitting flows of shear-thinning liquids in forward roll coating," AIChE J. 33, 741-746 (1987). Coyle, D. 1.. C. W. Macosko. and L. E. Scriven, "Stability of symmetric film splitting between counter-rotating cylinders," J. Fluid Mech. 216, 437-458 (1990). Dodge, 1. S., "Quantitative measures of levelling." J. Paint Technol. 44,72-78 (1972). Fuller, G. G., C. A. Cathey. B. Hubbard. and J. Zebrowski, "Extensional viscosity measurements for lowviscosity fluids," 1. Rheol. 31, 235-249 (1987). Glass. J. E.. "Dynamics of roll spatter and tracking-part III: Importance of extensional viscosities." 1. Coatings Technol. 50, 56-71 (978). Greener, J. and S. Middleman, "A theory of roll coating of viscous and viscoelastic fluids," Polyrn. Eng. Sci. 15, 1-10 (1975). Kang, Y. T., K. Y. Lee, and T. J. Liu, "The effect of polymer additives on the performance of a two-roll coater,' J. Appl. Polym. Sci. 43, 1187-1195 (1991). Kheshgi, H. K. and L. E. Scriven, "The evolution of disturbances in horizontal films," Chem. Eng. Sci. 43, 793-801 (1988) Krieger, I. M. and T, A, Huang, .. Axial flow through a narrow annulus, I. The kinetic energy correction," J. Rheo!. 38,141-[50 (1994). Krieger. I. M. and P. Mode, "Centering errors in annular flow," Xlth International Congress on Rheology, Brussels. Belgium (August, 1992). Kristiansen. K, "Rheological properties of paint systems during film formation," Ph.D, thesis, Technical University of Denmark (1991). Laun, H. M. and G. Hirsch. "New laboratory tests to measure rheological properties of paper coatings in transient and steady-state flows," Rheo!. Acta 28. 267-280 (1989), Matsuda, T. and W. H. Brendley, Jr. "Transfer and levelling processes and rheological design in water-borne reverse-roll coating systems," J, Coatings Techno!. 51,46-60 (1979). Mill, C. C. and G. R. South, "Formation of ribs on rotating rollers," J. Fluid Mech, 28, 523-529 (1967). Mode, P. {private communication. 19921. Orchard, S. E.. "On surface levelling in viscous liquids and gels," Appl. Sci. Res. Sec. A 11, 451-464 (1962). Pangalos, Goo J. M. Delay, and M. B. Lyne, "Rheological properties of new inks," 1. Rheol. 29, 471-491 (1985). Patton, T. Coo "Paint flow and pigment dispersion," Levelling (Wiley, New York, 1978), Chap. 28, pp. 551 -569. Pitts. E, and 1. Greiller, "The flow of thin liquid films between rollers," J. F[uid Mech. n. 33-50 (1961), Savage, M. D., "Mathematical models for coating processes," J. Fluid Mech. 117,443-455 (1982). Schunk, P. Roo 1. M. De Santos, and L. E. Scriven, "Flow of Newtonian liquids in opposed-nozzles configuration," J. Rheol. 34, 387-414 (l990l. Soules, D. A., R. H. Fernando, and J. E. Glass. "Dynamic uniaxial extension viscosity: Effects in roll application. I. Rib and web growth in commercial coatings," J. Rheol. 32, 181-198 (1988). Wallpott, Goo J, Ramthun, and L. Reif, "Prevision de la machinabilite sur les coucheuses lame trainante l'aide de rnesures rheologiques." Colloque A,R.S. Paul Neumann, Grenoble, France (September, [991). a a Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 193.48.255.141 On: Fri, 03 Jan 2014 11:09:01