Ind. Eng. Chem. Prod. Res. Dev. lB81, 20, 515-519
515
Nip Flow and Tack of Printing Inks Jan Mewls" and Frans Dobbels Department of Chemical Engineering, K. U. Leuven, de Croylaan 2, E3030 Leuven, Belgium
The squeezing flow of a printing ink in the nip between rotating rollers is investigated. A numerical analysis is used which takes into account the viscoelastic nonlinear behavior of the deformable roller. Contributions from the central creeping flow zone to tack reading are calculated and compared with experiments for three different rubbers on the measuring roller. The said zone only contributes a minor fraction to the tack force. Nevertheless the effect of roller characteristics on performance can be understood on the basis of the creeping flow theory. Slip is found to be an important element in quality considerations of tackmeters. Its effect on tack is investigated theoretically and experimentally. The flow pattern is so complicated that no simple scaling-up procedure can be formulated. Systematic experiments show a vehicle-related effect which might be associated with fluid elasticity.
Introduction Sets of rotating rollers, in contact with each other through a thin layer of liquid, occur frequently in industrial processes. Lubrication, calendering, and printing belong to the best known examples of this category. Often such a system of rollers is used to produce a liquid layer or sheet with constant thickness, as exemplified by roller coating or printing. In some cases the liquid which is being applied does not have a very high viscosity. It might be necessary then to replace some of the metal rollers by deformable rubber rollers, especially if an accurate and constant layer thickness is essential. These rollers are normally obtained by covering a metal core with a thick layer of natural or synthetic rubber. I t can be understood intuitively that the flow between a rigid and a deformable roller proceeds more smoothly than the flow between two rigid rollers. Unfortunately, the normal theoretical descriptions of printing inks in nip flow are based on the assumption of rigid rollers (Banks and Mill, 1954; Myers and Hoffman, 1961; Mill, 1967). This simplification was necessary because no suitable method was available to incorporate the nonlinear viscoelastic behavior of rubber in the analysis. Recently, the present authors developed a numerical method which made it possible to analyze flow in the nip between a rubber and a metal roller (Dobbels and Mewis, 1978). The results of this approach for the printing process and especially for the analysis of tackmeters are discussed here. I t is attempted to determine the most important material characteristics and to elucidate the physical meaning of tack. Description of Rubber Deformation in Rolling Contact Both natural and synthetic rubbers deform in a complex manner during rolling contact. In order to obtain a realistic picture of nip geometry it is essential to take into account the viscoelastic response of the rubber. Viscoelastic deformations in rolling contact have been dealt with by various authors, e.g., Morland (1962) and Harvey (1975). However, these authors apply a spectral description of material behavior which makes engineering calculations very difficult. In addition, nonlinearity is neglected in these analyses. At Darmstadt, Pfeiffer (1970) and Gluck (1976) have developed a nonlinear model which they have applied to dry rolling contact. They characterize the rubber in oscillatory deformations by a complex modulus E*, the magnitude of which depends on frequency (w) and 0196-4321/81/1220-0515$01.25/0
Table I. Material Characterization of Three Tackmeter Rollers ( w , = 2000 rad/s; e, = 50 pm) /E,,*/, shore roller a, MPa a2 80 cy hardness
1 2 3
-0.1 -0.1 -0.2
50.3 39.1 55.1
0.175 0.242 0.338
11'24' 15'36' 30"46'
0.381 0.358 0.308
70 60 70
amplitude (em). The loss tangent (tg 6) was found to depend only on frequency
The subscript 0 refers to arbitrary reference conditions. Five model parameters are required in the description of eq 1, but this is a minimum if one wants to include the nonlinear effects of frequency and amplitude. It can be shown (Dobbels and Mewis, 1978) that during nip flow the actual deformation of the rubber closely approximates a sine wave. Hence the given material description can be used adequately in nip flow calculations. It is assumed then that the deformation within the rubber is homogeneous and that there are no volume or thickness effects. In order to reduce the possible effects of these simplifications, the material parameters should be determined under measuring conditions as close as possible to the real situation. Here they are obtained by analyzing the rotation in dry contact between the rubber roller and a metal roller. As a function of the deformation amplitude e,, and the frequency of the sinusoidal deformation, the total normal force N and the driving moment T between the rollers are measured. From these values E* and tg 6 are calculated making use of the equations that relate T and N to the material parameters (Dobbels and Mewis, 1978). The calculations require a2 and a3 to be known. Hence an iteration procedure is used. The parameter for nonlinearity a1is found by selecting the value that gives the best fit of the IE*I-em data. Clearly, various other statistical procedures could be used as well. The characterization method described above is applied on three different rollers made to be used on a commercial tackmeter. The necessary experiments were performed at the T. H. Darmstadt, where a suitable device is available 0 1981 American Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20,No. 3, 1981
516
a
/?
....
.-
a
Figure 1. Experimental determination of dynamic roller characteristics at constant maximum deformation e,: a, 100 pm; b, 70 pm; c, 50 wm; roller: -, 1;--, 2; - - -,3; la: normal force per unit roller length; lb: driving moment per unit roller length.
Figure 2. Geometry of nip flow involving a deformable roller
(Gluck, 1976). As an illustration one set of data is shown in Figure 1. The total force N and the torque T are measured at preset levels of deformation, i.e., 50,70, and 100 pm, and at various speeds (nz)of the rigid roller. The selected values of e , are a compromise between the requirements of accuracy, for which high deformations are desirable, and the necessity to simulate closely the experimental conditions, which would require lower values (15-35 pm). Table I contains the calculated parameters for the three rollers. The material of roller 1and 2 is butyl rubber, whereas roller 3 is covered with nitrile rubber.
Comparison between Theory and Experiment The contact zone between rotating rollers can be conveniently divided in three zones (Figure 2). At the nip entrance some liquid is pushed back and forms a rotating mass in the so-called bank region. Here the necessary pressure is built up to push the liquid through the nip. In the central zone a region of regular creeping flow is assumed. The available pressure is used here to pump the material through the narrow channel between the rollers. Near the exit the separating rollers tend to create a negative pressure in the liquid. The latter causes the liquid to cavitate and to form filaments between the rollers. Only the creeping flow zone can be analyzed properly at this moment. A lubrication theory approximation provides shear rates and stress gradients either for rigid rollers or for rollers with a known deformation behavior. Some assumption has to be made about the limiting tensile stress
at the onset of cavitation in order to calculate the normal stress distribution by integration. A second integration provides the total normal force and global torque between the rotating rollers. If the tensile stresses near the trailing edge are small compared to the compressive stresses near the nip entrance, the contribution of the former to the total normal force can be neglected. In this case the creeping flow analysis can lead to good estimates of the global forces between the rollers. The limiting value for the negative pressure has been estimated to be about 106Pa for oils and resinous solutions in practical situations (Banks and Mill, 1954; Strasburger, 1958; Mill, 1967). In high-pressure applications this value can be set equal to zero without much loss of accuracy. High-pressure calculations have been published recently for nip flow with rubber rollers (Dobbels and Mewis, 1978). The results agree with available data for printing press simulations (Gluck, 1976). Hence it seems possible now to understand the effect of the rubber on printing ink flow during printing. From experience with tackmeters one could assume that the total force between printing press rollers is clearly affected by the flow of the printing ink. This turns out not to be the case (Dobbels and Mewis, 1978), a result which can be understood on the basis of rubber deformation. This discrepancy between tackmeter and printing press is explained by differences in pressure. On most commerical tackmeters the linear pressure is caused by weight of the measuring roller which is positioned on top of a metal central roller. Sometimes the weight is supplemented with a spring force. Here a Tack-O-Scope is used with a linear pressure of 100 N/m, clearly below linear pressures on a printing press. Tackmeters are used extensively to test printing inks. Hence it is important to understand the fluid mechanics on such instruments. If the numerical simulation of nip flow with a rubber roller is applied to the Tack-O-Scope,theoretical estimates for the extreme values of the stress profile can be obtained. It is found that peak pressures can be of the same order as the expected peak tensile stresses. Hence the latter cannot be neglected anymore, contrary to the case of the high-pressure applications. For the same reason cavitation is not expected to entail a similar reduction in tensile stresses as with high pressures. This can be seen readily by calculating the peak tensile stress assuming creeping flow throughout the whole nip. Rigid roller theories have led to the conclusion that the central creeping flow region provides the major contribution to the drag force which is used as a measure of tack (Mill, 1961,1967). The creeping flow region is then defined as the central zone in which the local distance between the rollers is smaller than the sum of the layer thicknesses of liquid on the rollers outside the nip. Our simulation method was applied to the Tack-O-Scope for the three rollers of Table I with a Newtonian oil (a = 10.2 Pass). The calculated contribution of the central creeping flow region is compared with the measured tack or drag force D in Figure 3. On the basis of the present analysis it is concluded that the central zone contributes in this case about 15% of the measured drag force. This conclusion is corroborated by earlier experiments that showed an important effect of ambient pressure on tack (Zettlemoyer and Myers, 1960). The latter result is compatible with a small contribution from the central zone. A closer look at the calculations shows that the tensile stresses at the end of the central region are of the order of lo4 Pa, clearly below the critical values for cavitation as given above. In order to verify that the present analysis
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 3, 1981 517
,a
-
__--
-------
-
50
loo
n(m/min)
zoo
Figure 3. Calculated contributions of the central creeping flow zone (lower curves) to the measured tack values (upper curves) (1 TOS unit = 0.0533 N/m) (symbols, see Figure 1).
is realistic, the creeping flow region is assumed to continue beyond the central region as defined above. I t is then calculated how far the zone has to be extended to find the measured values of tack. For Newtonian oils with viscosities of 5.8 and 10.2 Pa.s, the necessary expansion of nip width turns out to be 20-25%. The corresponding limiting tensile stresses amount to 200-600 kPa. Both the values for nip expansion and limiting stress seem reasonable. These calculations indicate that a minor contribution to tack from the central zone is indeed compatible with acceptable stress distributions in a nip with a rubber roller. I t is therefore concluded that tack is mainly determined by the flow near the nip exit, at least for the instrument under consideration. As with previous calculations, the present one does not provide an a priori value for tack, owing to the interference of cavitation. Nevertheless one could hope that the simulation of the roller deformation would contribute to a better understanding of tack measurement and printing ink flow. Figure 3 indicates that, notwithstanding the large absolute discrepancy, the relative position of the calculated curves is in agreement with the measured behavior. To a first approximation the cavitation zone only seems to shift the curves. By perturbation of the material parameters it can be seen which of them are most important in tack measurement. The calculations learn that the loss tangent tg 6 has the greatest relative effect. It can be verified in Figure 3 that both the calculations and the experiments order the rollers according to their value of tg 6. The latter determines the position in the nip where the pressure reaches a peak value and also changes the level of this peak. Consequently, it can alter considerably the moment of the pressure profile and therefore also the drag force, as indicated below. At the same time it is concluded that simple measures of roller hardness, as Shore hardness, are unsuitable for characterizing rollers. They do not provide any information on the damping properties which are expressed through tg 6. The previous discussion as well as Figure 3 refers to Newtonian liquids. In printing ink technology one is primarily interested in non-Newtonian materials. The numerical simulation has not been extended yet to such fluids. However, from rigid roller theories one can expect the same arguments to be basically balid for more complicated fluids (Mewis and Verbist, 1970). This is verified here by measuring various printing inks with the rollers of Table I. The results are illustrated by the data obtained on one printing ink. The rheogram of this material is represented in Figure 4. Again tack is measured on Tack-0-Scope; the printing ink volume is 0.5 cm3.
Figure 4. Rheogram of the printing ink used in comparative tack measurements, Table 11. Table 11. Effect of Rubber o n Tack and Slip Curves of a Printing Ink 1 2 3
188 201 241
0.566 0.551 0.521
0.80 1.08 1.42
-0.240 -0.192 -0.128
The tack (D)vs. speed (n)curves are described by a power law relation
D
= Do(
:yl
where subscript 0 refers to an arbitrary reference condition; here no is taken as 200 m/min. The corresponding parameters of the experimental tack curves are given in Table 11. This table also contains slip data which will be discussed later. A comparison between Tables I and I1 shows that the relative magnitude of tack and the relative change with roller speed follow the corresponding values for the loss tangent of the rubbers. Other printing inks give similar results. Hence the conclusion that the rubber affects the tack reading basically through its loss tangent is confirmed. This result can be used in several ways. From comparative measurements with the same liquid and different rollers the relative damping properties of the roller material can be estimated. Further the result provides a better insight in the physical meanining of tack itself and of the slope of tack-velocity curves. A final remark concerns the scaling-up of laboratory experiments. The analysis indicates that, if the defomation of the rollers is taken into account, the shear rate and stress distribution in the nip changes in a complicated manner with variables as viscosity, layer thickness, pressure, and roller material (Dobbels and Mewis, 1978). In addition, it has been shown above that a variation in balance between compressive and tensile stresses drastically changes the effect of fluid properties on the global forces. Hence a simple scaling-up procedure for the kinematics from a tackmeter to a printing press seems impossible. A tack measurement provides valuable information about a printing ink, but there is no easy mathematical link with overall press behavior, even for nip flow. Effect of Slip and Roller Bearing Friction As indicated earlier, the numerical analysis of the high-pressure case suggests only a minor effect of the liquid properties on the resulting drag force between the rotating rollers. This analysis has been used to screen for process parameters that are more sensitive to the liquid properties. The system under consideration consisted of an externally
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driven roller and a second roller dragged along through the liquid in the nip. The viscous drag is accompanied by a certain slippage between the rollers. The slippage is expressed as the relative difference between the linear speeds at the roller surfaces, (nz- n,)/nl, or as a percentage by multiplying the previous value by 100. According to the computations, slip seems to be one of the very few parameters that is sensitive to liquid properties in highpressure nip flow. The question now arises how important and how variable slip is in low-pressure applications, especially for tackmeters. Rigid roller theories indicate that slip changes inversely proportional with velocity if the normal force between the rollers ramins constant. The latter is the case for tackmeters. With a deformable roller the roller characteristics are expected to interfere with the slip. A computation based on the usual high-pressure assumptions confirms this supposition. In particular the loss tangent is found again to be the most important rubber parameter. Logically, the effect is more pronounced at small layer thicknesses than at large ones. With increasing film thickness the case of immersed rigid roller can be approached. Slip does not only depend on roller characteristics and liquid properties. Obviously the friction in the roller bearings will interfere as well, thus affecting the tack reading. Therefore this factor is also included in the present investigation. The interaction between slip and the other forces follows from the torque balance around the axis of the measuring roller. This balance requires an equilibrium between the total moment of the pressure profile in the nip (M,), the moment of the drag forces D acting on the surface of the measuring roller (radius R ) , and the moment of the friction forces in the bearing ( M J M , + D-R + M f= 0 (3) In the absence of friction the drag forces should balance the moment of the pressure profile. This would normally cause a minor slip between the rollers. The presence of friction in the bearings distorts the balance between D.R and M,, causing an increase in slip and drag force. It is not the purpose of a tackmeter to measure bearing friction. Consequently, it seems logical to keep the contribution of friction to drag force small. Otherwise the measurement would still provide some information about the liquid although of a different nature. The Tack-O-Scope has been modified in order to measure the relative difference in linear velocity between the rollers. A circular disk is mounted on each roller axis. Near the rim, holes are drilled uniformly distributed along the circumference of the disks. Their number is proportional to the diameter of the other roller. By means of light sources and photoelectric cells two pulsed signals are generated, the frequency of which is proportional to the speed of the respective rollers. The pulses are amplified and the slip percentage calculated on a Monsanto Counter-Timer 101B. For the printing ink of Figure 4 slip (3)-velocity ( n ) curves were measured as well as tack curves, both under identical conditions. The slip curves are expressed as power law relations s =
);*
(4)
The parameters so and Pz for the three rubbers of Table I are given in Table 11. As with tack, the slip parameters follow the relative changes in tg 6 and its velocity dependence. Again the theoretical predictions are qualitatively confirmed by the experiments. Therefore it can be con-
lop-----
SO
07-
100
n lmlmrnl
zoo
50
200
100
n lmlmrnl
Figure 5. Tack and slip measurements for five printing inks, using one measuring roller.
cluded that the model of eq 1 for the rubber behavior of the rollers, combined with the numerical analysis used, provide a suitable way to approach the effect of rubber in nip flow. Up to now various rubbers have been discussed using either Newtonian liquids or a single printing ink (Figure 4). Tackmeters are normally used with one set of rollers for the purpose of discriminating between different printing inks. Figure 5 shows tack and slip curves for the normal Tack-O-Scope roller and various inks. Whereas slip and tack change both in the same direction if the roller properties are changed, the changes are normally in the opposite direction if the liquid properties are changed. It is to be expected that tack increases and slip decreases if the liquid viscosity is increased. The relative difference in slip between various liquids can be considerable as could be expected theoretically. However, these differences are not amplified under low-pressure conditions as is the case for the drag force. Furthermore, Figure 5 suggests a more complex effect of liquid properties than predicted with the present theories. With a value for the slip available, its contribution to the measured tack can be estimated within the assumptions of the theoretical analysis. Calculations for the present tackmeter indicate that slip contributes rarely more than 5 % to the total tack. Obviously this value is linked to the type of roller bearings and to the nature of the rollers and the condition of the individual instrument. It cannot be extrapolated to other types of tackmeters. Mill (1961) has found slip values of about 10% on his tackmeter, much larger than the ones reported here. In that particular case the contribution of slip to tack has not been calculated. It will be examined how an increase in bearing friction will affect the tack reading. For this purpose, bearing friction is increased artificially by mounting a sliding contact on the axis of the measuring roller. Roller bearing friction itself cannot be readily measured on tackmeters. Therefore the resulting change in slip will be used as its measure instead. Figure 6 contains the resulting tack and slip curves. The slip has been kept below 4 % . Nevertheless there is a drastic change in the tack curves. The average readings increase considerably and the slopes decrease because the slip contribution is relatively more important at lower speeds. From Figure 6 it is concluded that slip and bearing friction are relevant parameters in tack interpretation, tackmeter quality assessment, and tackmeter conversion. Hence slip measurements, especially for low viscosity liquids, constitute a suitable tool in tackmeter assessment.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 20, No. 3, 1981 510 600 -
6
2.2
1
I
I
I
1
100
1
70 L 50
100
n imlminl
Figure 6. Tack and slip measurements with increased bearing friction: 0,tack values; X, slip; -, normal bearing friction; -- and - - -,increased bearing friction.
Tack and Viscoelasticity of Printing I n k s Figure 5 suggests that more than a single rheological parameter of the liquid is required to describe tack adequately. Rigid roller theories (Mewis and Verbist, 1970) indicate that taking into account shear thinning behavior does not solve the problem. This is demonstrated in the following experiment. Three homogeneous liquids, a mineral oil (A), stand oil (B), and an alkyd varnish (C) are pigmented at several levels between 0 and 10% pigment volume concentration. The pigments include TiOz (for C), benzidine yellow (for B and C), and carbon black (for A, B, and C). The pigmentation shifts the tack curves in a way which can be understood (Hellinckx and Mewis, 1969) on the basis of viscosity changes. However the vehicle itself has an identifiable effect. This can be seen if the tack readings are divided by the reading at a reference speed, say 50 m/min, to show the relative changes of tack with speed (Figure 7). In such a plot the curves cluster together according to the type of vehicle. The viscosity of the dispersion changes in each group with pigment concentration. Hence this characteristic cannot be responsible for the shape of the curves. One possible parameter might be the viscoelasticity of the vehicle. The mineral oil is nearly Newtonian. I t is known that dispersions in mineral oil can give rather steep tack-velocity curves. The other two vehicles are definitely viscoelastic. If measured at high frequencies (45 kHz) on a travelling wave device (Hellinckx and Mewis, 1969), they show a loss tangent of about 3.0. The data could then suggest that viscoelasticity of the vehicle tends to flatten the tack-speed curves. The difference between B and C is not explained in this manner. It could be argued that the viscoelasticity measurements are not performed in the correct frequency range and that nip flow is more complex than oscillatory flow. Exploratory measurements on nonpigmented fluids confirmed the present result: inelastic liquids give larger slopes. However, no elastic characteristic could be found that correlated with the data. Conclusions The theory of tackmeters has been extended to take into account the nonlinear viscoelastic behavior of the meas-
1
/I
7’ zoo7
I
I
I
1.0s!?’
rba
2b0
2ko ~ n (m/minl
o
’
Figure 7. Tack values relative to the value at 50 m/min for three homogeneous liquids containing various levels of pigmentation (vertical line segments indicate range of scatter of data): A, mineral oil; B, stand oil; C, alkyd varnish.
uring roller. Contrary to the case of nip flow on a printing press, the central creeping flow zone contributes only a minor fraction to the drag force between the rollers of a tackmeter. Nevertheless, tack as well as slip between the rollers follows qualitatively the trends computed from the central zone contribution under changes of roller properties. Hence it is possible to predict qualitatively the effect of the measuring roller material on tack readings and on the slip between the rollers. It is found that the loss tangent of the rubber and its frequency dependence are the most important characteristics of the rubber. To a first approximation slip follows the predicted changes with liquid properties, but the detailed pattern is more complex. Viscoelasticity has been identified as possibly an additional liquid property which affects tack. In particular, elasticity of the liquid seems to flatten the tack-speed curves. Bearing friction is found to be an important factor in tack measurement. It has a pronounced effect on the flow in the nip. Hence tack curves are changed also, both the level and the slope. For high sensitivity a low bearing friction is recommended. Acknowledgment The help of Professor Scheuter of the T. H. Darmstadt in providing the facilities to characterize the rubber rollers used in the present experiments is gladly acknowledged. Literature Cited Banks, W. H.; MIII, C. C. Proc. R. SOC. London Ser. A 1954, 223, 414. Dobbels, F.; Mewis, J. Chem. Eng. Sci. 1978, 33, 493. Douglas, A. F.; Lewis, 0. A.; Spaull, A. J. B. Rheol. Acta 1971, 70, 382. Gliick, M. Ph.D. Thesis, T. H. Darmstadt, 1976. Harvey, R. 0. Quart. J . Mech. Appl. Math. 1975, 10(1), 1. HelHnckx, L.; Mewis, J. Rheol. Acta 1969, 8 , 319. Mewis, J.; Verblst, R. “X FATIPEC Congress”, Verlag Chemle: Weinhelm, 0. F. R., 1970; p 591. Mill, C. C. J . 011 Colour Chem. Assoc. 1961, 44, 596. Mill, C. C. J . OilCobur Chem. Assoc. 1967, 50, 398. Morland, L. W. J. Appl. Mech. 1982, 29, 345. Myers, R. R.; Hoffman, R. D. Trans. Soc. Rheol. 1981, 5, 317. Pfelffer, G. Ph.D. Thesis, T. H. Darmstadt, 1970. Strassburger, H. J. C01bid Sci. 1958, 13, 218. Zettlemoyer, A. C.; Myers, R. R. “Rheology Theory and Applications”, Elrlch, F. R., Ed.; Vol. 111, Academic Press: New York, 1960; p 145.
Received for review October 15, 1980 Accepted April 8, 1981
