Slip Effects in Capillary and Parallel Disk Torsional Flows of Highly Filled Suspensions

Transcription

1 Slip Effects in Capillary and Parallel Disk Torsional Flows of Highly Filled Suspensions ULKU YILMAZER and DILHAN M. KALYON,* Department of Chemistry and Chemical Engineering, Stevens Institute of Technology, Castle Point, Hoboken, New Jersey Synopsis The shear viscosity material function of a highly filled suspension consisting of a Newtonian poly(butadiene acrylonitrile acrylic acid terpolymer) matrix, PBAN, mixed with an ammonium sulfate tiller at 60% by volume was studied. Both capillary and parallel disk torsional flows were employed. The rheological characterization revealed strong slip of the suspension at the walls over a broad range of shear stresses in both types of flows. The slip velocity increased approximately linearly with the shear stress. In capillary flows, above a critical shear stress, flow took place in a pluglike manner, owing to slip at the wall. The experimental findings were further elucidated to determine the slip layer thickness and the apparent shear viscosity behavior of highly filled suspensions at high shear stress at the wall values. It was concluded that the slip effects dominate the flow of highly filled suspensions and the true flow and deformation characteristics of the highly filled suspensions may be overshadowed by slip at the walls, INTRODUCTION Highly filled suspensions, which are filled close to their maximum packing fraction, are processed widely in various industries. They include the batch and continuous compounding and shaping of thermoplastic and thermosetting resins, molding of ceramic articles, and mixing of solid fuels. The rheological behavior of dilute as well as concentrated suspensions have been the subject of several reviews. -7 A detailed understanding of the rheology of highly filled systems is a prerequisite for their optimum and safe processing. However, the characterization of these suspensions is first complicated by structural changes, which may occur during the characterization, and second by possible slip at the walls of the viscometers. Wall slip effects in capillary flows of low to moderately concentrated suspensions of particulates in low viscosity matrices have been reported.g- 6 A recent articlen reviews the slip phenomenon *To whom correspondence should be addressed by The Society of Rheology, Inc. Published by John Wiley & Sons, Inc. Journal of Rheology, 33(81, (1989) CCC 0148~6055/89/ $04.00

2 1198 YILMAZER AND KALYON in suspensions as well as in polymer solutions. Wall slip in suspensions is closely related to the migration effects encountered in liquids containing very little 8* s or moderate amounts of particulates. It has been found a~ that during the flow of very dilute suspensions in a capillary, particles move away from the wall and the center, to a distance of approximately 0.6 radii from the center. Owing to the migration effect, it has also been observed that as a moderately concentrated suspension of rigid spheres flows from a large reservoir through a narrow tube, a reduction in the concentration of spheres takes place. In the capillary flow of polymer melts, containing moderate concentrations of particulate fillers, migration effects increase with increasing shear stress.21a22 In this report, the rheological behavior of a highly filled (60% by volume) suspension was studied employing capillary and parallel disk torsional flows with emphasis on the wall slip phenomenon. Information on the pressure as well as the drag flow behavior of such highly concentrated suspensions with highly viscous matrices had been lacking. This study is part of a larger study, which includes the simulation and experimental studies of continuous processing of highly filled suspensions in corotating twin screw extruders MATERIALS The matrix used in this study was poly(butadiene acrylonitrile acrylic acid) terpolymer, PBAN, manufactured by American Rubber Company [PBAN, HP terpolymer]. Its specific gravity is It is a Newtonian fluid with a viscosity of 37 Pa-s at 25 C. The filler was a FCC extra fine grade ammonium sulfate supplied by Delta Chemicals. It was ground to the particle size distribution shown in Table I. The particles, as shown in the scanning electron micrograph of Figure 1, exhibit low aspect ratios. Ammonium sulfate has a specific gravity of The suspension was prepared in a Baker Perkins 50.8 mm, clam shell design, fully intermeshing, corotating twin screw extrnder. The volume fraction of the tiller in the suspension was 60 t0.2% by volume (verified independently by solvent extraction through adaptation of ASTM procedure D494-72). EXPERIMENTAL The samples compounded in the twin screw extruder were characterized employing capillary and parallel disk torsional

3 HIGHLY CONCENTRATED SUSPENSIONS 1199 TABLE I Particle Size Distribution of Ammonium Sulfate Used in This Study Characteristic length (pm) Mean = 23 pm. SD = 13 Wm. Percentage of particles flows at 25 C. For capillary flow experiments, an Instron Capillary Rheometer, Model TFD, was employed in conjunction with three sets of capillaries. Each set contained four capillaries. In each set, the capillaries had the same diameter but differed in their length over diameter ratios. The dimensions of the capillaries are shown in Table II. The three sets of capillaries had the diameters of 1.32, 1.59, and 1.98 mm (ko.02 mm). The length over diameter ratios varied between zero and The materials of this study and the loading level were selected to avoid unstable flows associated with mat formation and filter- Fig. 1. study. Scanning electron micrograph of the ammonium sulfate used in this

4 1200 YILMAZER AND KALYON Diameter TABLE II Dimensions of Dies Used in CaDillarv Flow Measurement (mm) (LID) = 0 Length (mm) D, = Dz = Da = ing of the matrix. The data were collected by repeating the experiment at least three times per run, followed by statistical analysis. In order to investigate the rheological behavior at lower shear stresses, the suspension was characterized employing a Rheometrics Mechanical Spectrometer, Model RMS 800, in the parallel disk configuration. Steady shear data were obtained by using various gap heights in the range of 1 to 4 mm. New samples were used in each run. BACKGROUND Capillary Flow In capillary flows the shear stress at the wall, r,, can be determined by correcting for the pressure losses associated with the end effects.25 It is given by: APD 7, = 4(L + ND) (1) where AP is the total pressure drop over the capillary, L is the length and D is the diameter of the capillary, and N is the equivalent length associated with the end correction.25 The equivalent length can be determined by the extrapolation of the pressure drop versus the length over diameter ratio curve to intersect with the length over diameter axis.25 The apparent shear rate, +,, is determined from Sv/D, where v is the average fluid velocity of the fluid, and the apparent viscosity, Q, can be calculated from: To determine the slip effects, capillaries with the same length over diameter ratios but with different diameters are usedez6 The (2)

5 HIGHLY CONCENTRATED SUSPENSIONS 1201 analysis proposed by Mooneyz6 for fully developed, incompressible, isothermal, and laminar flow in circular tubes with a slip velocity of u, at the wall yields: 8u -=- T2i,clT i- s D (3) where G- is the shear stress and Jo is the true shear rate. Differentiating the last equation with respect to l/d at constant shear stress at the wall, TV, one W/D) = 8u, TW Thus, the plot of the apparent shear rate, (8V/D) = Jo,, versus l/d at constant rw should give a straight line with a slope of au,. The contribution of slip to the total volumetric flow rate can be found as follows. The flow rate due to slip, Q,, is given by: Q, = (d4)d2u, (5) Thus, ratio of flow due to slip, Q,, over the total flow rate, Q, is given as: t&,/q) = Wff) (6) Parallel Disk Torsional Flow In order to determine the slip velocity and the actual shear rate in parallel disk experiments Yoshimura and Prud hommez7 outlined a method based on performing two sets of experiments at two gap heights and a procedure for correcting the parallel disk torsional flow data. The following method is a generalization of that mentioned procedure. It provides a better accuracy, since the data arising from more than two gap heights are utilized in our procedure. In the parallel disk torsional flow, the apparent shear rate, $J~, (not corrected for slip effects) is a linear function of the radius, r, given by: where r is the radial distance from the center of the disk, H is the gap height, and Cl is the angular velocity of the upper disk relative to the lower one. The apparent shear rate is related to (7)

6 1202 YILMAZER AND KALYON the true shear rate, +, and the slip velocity, equation:27 u,, by the following 9, = $7) + y (8) Here j(~) and U,(T) that is, the true shear rate and the slip velocity, respectively, are functions of the shear stress, r = ~=o where z and 8 are the axial and angular components of the cylindrical coordinate system. The shear stress at the edge of the disk, TR, can be determined from: TR=j-$ 3+ [ &EE] where T is the torque required to rotate the upper disk and qar is the apparent shear rate at the edge of the disk obtained by substituting r = R in Eq. (7). Equation (9) is similar to the Rabinowitsch28 correction used in the correction of capillary data. In Equation (9) the function f = d(ln T)/d(ln jaa) is dependent on the gap height used, as previously pointed out.27 Equation (8) also applies at r = R and thus it can be written as +df =?R(~R) + 247R) H (9) (10) The last equation shows that if plots of jar versus l/h are drawn at constant rs, then straight lines are obtained. The extrapolated intercepts are equal to jr(rj, that is, the true shear rate at the edge, and the slopes are equal to 2u,(~~). RESULTS AND DISCUSSIONS Capillary Flow Data Figure 2 shows the end effect corrected apparent shear viscosity versus the apparent shear rate behavior of the suspension with the series of capillaries that have a length over diameter (L/D) ratio of The values of the equivalent length, N, were determined using the Bagley correction.25 They varied between 2 and 22 and were generally greater than those commonly observed in polymer melts. This emphasizes the importance of the corrections for the end effects. Similarly, in slip studies of ge1s,17r2g it was found that, in the presence of slip, end effects were very large and should not be overlooked. From Figure 2 it is seen that in the 10~ apparent shear rate range the apparent shear viscos-

7 HIGHLY CONCENTRATED SUSPENSIONS 1203 a5ooo- * 2 2. pm * l = A A -r--a----.m >. l i%= D-l.Q8lllm A D-1.5Qmm. D=l.32mm. Equation (IQ) LID-57.6 Lo. --- A--C- z 2 cd 3 < loo0,,,,,, IO 30 ioil Apparent Shear Rate, i/s Fig. 2. The apparent shear viscosity versus the apparent shear rate. The L/D ratio is 57.6 for all dies. ity decreases with the increasing apparent shear rate, but levels off at higher apparent shear rates. However, the apparent shear viscosity values obtained employing various capillaries with the same L/D ratio, but with different diameters do not overlap. The apparent shear viscosity values measured with capillaries that have smaller diameters are smaller. This indicates that wall slip has taken place. Other sets of capillary data arising from the use of capillaries which have L/D of 19.2 and 38.4 exhibited similar effects of diameter dependence. Thus, the data need to be corrected for slip effects as well as for end effects. The slip effect was analyzed as follows: The end effect corrected shear stress at the wall of the capillary versus the apparent shear rate data was plotted as shown in Figure 3. Then ya was read at constant ru for each diameter and 9, versus l/d was plotted at constant T, as implied by Eq. 4. Plots of 9, versus l/d at constant 7, are shown in Figure 4 for the data set pertaining to the length over diameter ratio of The data points fall reasonably well on the linear regression curves. Other sets of capillaries with other L/D ratios also give rise to linear plots of 9, versus l/d. The slip velocities calculated from the slopes of the lines are shown in Figure 5 along with data obtained from dies that have the L/D of 19.2 and Data arising from parallel disk torsional flow are also included in the figure and will be

8 1204 YILMAZER AND KALYON D-l.S8mm * D. 1.59mn. D- 1.3Zrrun. L/D-67.6 mol Apparent Shear Rate, l/s Fig. 3. The shear stress at the wall versus the apparent shear rate. The L/D ratio is 57.6 for all dies Pa -.A Pa --g Pa -El Pa A Pa Pa -e Pa --, Pa -+- it i/d, l/mm Fig. 4. The apparent shear rate versus l/d. The L/D ratio is The shear stress values are indicated.

9 HIGHLY CONCENTRATED SUSPENSIONS f :: >.cl m 0.01 L/D- IQ.2 0 LfDm UD.57.6 n Parallel Disk * Equation (12) 3N!atll ( 13) 1ca Shear Stress vxno, Pa 1WDOD Fig. 5. The slip velocity, u,, versus the shear stress, 7, or Q. The prediction of Eqs. (12) and (13) are shown. For the capillary data [Eq. (1211 a = 9.2 x lw5 (mm/pa s), for the combined data of the capillary and the parallel disk torsional flows (Equation 13) a = 2.34 x 10e5 mm/s(pa) and m = discussed later. The slip velocity versus the shear stress at the wall behavior obtained by capillary flow experiments can be described very well by the following equation. u, = ar, (12) The only parameter necessary, a, was determined as 9.2 X 10m5 mm/(pa-s). The best fit of Eq. (12) is shown in Figure 5. Various other forms of shear stress and slip velocity relationships were suggested for polymeric solutionsn and include: u, = a rz (13) where a and I?E are material parameters. Next?a versus l/d data appearing in Figure 4 were used, along with the corresponding?, versus l/d data for L/D of 19.2 and 38.4, in order to determine Q,/Q as a function of 7,. These results, obtained by employing Eq. (6), are shown in Figure 6, for capillaries of diameter 1.98 mm. In this figure, the data pertaining to all L/D ratios are reported. The 90% confidence intervals indicated on the figure were determined according to Student s t-distribution.

10 YILMAZER AND KALYON u ', / 0.4 r 0 T- 0.2 i- 0 ***I I A 5ooo loo00 1OOOOO 2mOOO Shear Stress, Pa Fig. 6. The ratio of (C&/Q) versus the shear stress at the wall for all L/D ratios in the capillary flow. The die diameter is 1.96 mm. Values of Q,/Q exceeding one were also reported by Jiang et al. for flow of a gel in capillaries. This was attributed to the presence of experimental errors, especially due to the end effects, which we also agree with. These errors are larger for capillaries of small length over diameter ratios. For example, when the data shown in Figure 6 were elucidated further, it was observed that the values of Q,/Q greater than one at the small shear stress values arise from dies that have L/D of 19.2 and It is thus interesting to review again the capillary data obtained with the high length over diameter ratio (i.e., L/D = 57.6) where errors associated with end corrections should be minimal. The values of the ratio Q,/Q for the capillaries with L/D of 57.6 are shown in Figure 7, indicating that the contribution of slip to the flow rate is relatively independent of the diameter, in the diameter range considered here. This figure also indicates that Q,/Q values increase with increasing shear stress. Furthermore, flow takes place in a pluglike manner, owing to slip, above a critical shear stress, T,,, CT,, = 40,000 Pa for the suspension of this study). The values of Q,/Q are approximately one for shear stresses at the wall, which are greater than 40,000 Pa. Above this stress, negligible deformation rates exist in the suspension.

11 HIGHLY CONCENTRATED SUSPENSIONS i-++9$-gm ii Fi D n D.l.98mm n D= 1.5hm l D=1.32mm 0 LID-57.6 m IDDDO 2cGoo 5DDoo 1DDcal PMXMO Shear Stress, Pa Fig. 7. The ratio of (Q,/Q) versus the shear stress at the wall for all capillary diameters. The L/D ratio is Parallel Disk Torsional Flow Data Figure 8 shows the shear stress at the edge, rr, versus the apparent shear rate, +&, behavior obtained with the parallel disk Gap Helghl - 2 mm GapWgM-3mm GapHelgM-4mm &xjl O.cQl Fig. 6. Shear stress, TV, versus the apparent shear rate, jar, (points) or the corrected shear rate, R, (asterisks and the continuous curve) in the parallel disk torsional flow measurements.

12 1208 YILMAZER AND KALYON configuration. The shear stress values obtained by employing various gap heights are different. This again indicates that slip has taken place also in the smaller shear stress range of the parallel disk torsional flow experiments. It was not possible to continue the parallel disk measurements above the shear rate range shown in Figure 8, since above this range the samples showed visible signs of fracture during steady flow. In Figure 9, the apparent shear rate versus the reciprocal height data at constant 7R are shown. The data points fall reasonably well on the regression lines drawn through the points confirming the validity of Eq. (10). The shear rates at radius R corrected for slip, &(T~), were then determined from the intercepts. In Figure 8, the shear stress TV, is also shown against -j+. Thus, the effect of slip correction is mainly to shift the shear stress values to smaller shear rates. The slip velocity values in parallel disk torsional flow experiments calculated using the slopes as implied by Eq. (10) are shown in Figure 5, together with the slip velocity values calculated from the capillary flow experiments. To our knowledge, these are the first slip data obtained by using two different methods of measurements covering several decades of shear stress. In spite of the differences in the surface characteristics of the rheometers, the data arising from the two experiments agreed well. It could 15OPa * 200 Pa I, :; l/h, l/mm 4W Pa 800 Pa 800 Pa Fig. 9. Apparent shear rate, jar, versus l/h in the parallel disk torsional flow experiments. The shear stress values are indicated.

13 HIGHLY CONCENTRATED SUSPENSIONS 1209 have been expected that the slip velocity would depend on the roughness of the surfaces of the viscometers. However, this dependence is not straightforward and also requires the consideration of suspension characteristics and the slip mechanism. If a resin-rich layer establishes near the wall, and covers the rough surface of the viscometer, the roughness of the surface may not play a significant role. However, if the liquid-rich layer is not thick enough to cover the surface roughness, slip may not take place. In Figure 5, the regression fit of the power law type of relation given in Eq. (13) is also shown. The value of a was determined as 2.34 x 10e5 mm/s(pa). 3 and m was found to be equal to Thus, it seems that the slip velocity increases with a slightly higher power of the shear stress at low shear stresses. However, within the accuracy of the experiments, a linear relation between the slip velocity and the shear stress can also represent the combined data equally well. Slip Layer Thickness The slip-aided flow of the suspension takes place by either true slip at the wall or by apparent slip, 712g*30 through the formation of a thin, liquid-rich layer at the wall with thickness 6 allowing the suspension to slide through. This layer is referred to as the slip layer thickness in the context of the apparent slip mechanism. 17C2g,30 The slip layer thickness, 6, can be derived on the basis of assuming the occurrence of a fully developed, twophase flow in the capillary, with a Newtonian, incompressible slip layer flowing under isothermal, laminar conditions at the capillary wall. No assumptions are made on the rheological properties of the suspension flowing in the core region. The boundary conditions used in solving the equation of motion are as follows: 1. There is no slip at the wall (for the slip layer) 2. The shear stress is continuous at the slip layer-core region interface 3. The velocity gradient is zero at the center of the capillary The result written for the slip velocity, u,, is:

14 1210 YILMAZER AND KALYON where q7g is the shear viscosity of the slip layer and D is the diameter of the capillary. Assuming that the slip layer thickness over the capillary diameter ratio, S/D, is much smaller than one, the slip layer thickness becomes: (15) Equation (15) was also derived in Jiang et al.: assuming the shear stress to be constant in the thin slip layer. In our experiments, it can be assumed that the slip layer consists of the matrix, PBAN, alone. In conjunction with the linear dependence of the slip velocity, u,, on the shear stress at the wall, as given by Eq. (12), the slip layer thickness can be obtained from Eq. (15) as: 6 = a7), (16) The slip layer thickness at high shear stress values is thus independent of the shear stress. Substituting 37 Pa-s (at the temperature used in our experiments, 25 C) for the viscosity of PBAN and 9.2 x 10m5 mm/(pa-s) for the parameter a, as determined earlier, the slip layer thickness is determined as 3.4 pm. Considering that the mean characteristic length of the particles in our experiments was 23 pm, the estimated slip layer thickness is reasonable. Plug Flow at High Shear Stresses The ramifications of the findings to the apparent shear viscosity versus the apparent shear rate behavior are as follows. At high shear stress at the wall values in capillary flows, the values of Q,/Q reach one, implying that the suspension flows as a plug. In this case the apparent shear rats is given by: &g=% (17) Substituting the slip velocity, u,, from Eq. (12) and rearranging we get: 09, 7, = - 8a (18) Thus, at high shear stress at the wall values (or high apparent shear rates), where plug flow occurs, the apparent shear viscosity would be given by D 7)a = G (19)

15 HIGHLY CONCENTRATED SUSPENSIONS 1211 In this case, the apparent shear viscosity would be independent of the apparent shear rate but directly proportional to the capillary diameter. This behavior is clearly shown to be true in Figure 2, where the apparent shear viscosity values are plotted versus the apparent shear rate for various diameters. Plateau values determined on the basis of Eq. (191, are indeed approached at high apparent shear rates where Eqs. (17)-(19) are valid. CONCLUSIONS The flow behavior of a highly filled PBAN/ammonium sulfate suspension was characterized employing parallel disk torsional and capillary viscometry. The flow of this highly filled suspension was found to be strongly affected by slip at the wall. The slip velocity at the wall values were determined over a wide range of shear stress, covering 100 to 200,000 Pa. The slip velocity was found to vary linearly with the shear stress, at high values of the shear stress at the wall. Upon reaching a critical shear stress at the wall, the capillary flow took place by the pluglike motion of the suspension through apparent slip at the wall. The slip layer thickness was calculated as 3.4 pm. The plug flow, which took place at high shear stress at the wall values, gives rise to an apparent shear rate-independent (but diameter-dependent1 apparent shear viscosity behavior in that range. Overall, this study emphasizes the necessity of employing laborious procedures to determine the true flow and deformation behavior of highly filled suspensions, which is dominated by slip at the wall effects. The presented data should have strong ramifications in the processing of such highly filled suspensions. We gratefully acknowledge the support of the Office of Naval Research under Grant Number N Some of the experimental work was carried out by Ms. B. Aral and Mr. J. Cherian of Stevens Institute of Technology. References 1. J. Mewis and A. J. B. Spaull, Adu. Colloid Interface Sci., 6, 173 ( G. V. Vinogradov and A. Y. Malkin, Rheology of Polymers, Mir Publishers, Moscow, 1980, Chap J. Mewis, Proc. Int. Cong. Rheol., 8th, 1, 149 (1980). 4. S. Onogi and T. Matsumoto, Polym. Eng. Rev., 1, 45 (1981). 5. M. R. Kamal and A. Mute], J. Polym. Eng., 5, 293 (1985). 6. A. B. Metzner, J. Rheol., 29, 739 (1985). 7. S.A. Khan and R. K. Prud homme, Reu. Chem. Eng., 3,205 (1987).

16 1212 YILMAZER AND KALYON 8. U. Yilmazer, C. Gogos, and D. Kalyon, Polymer Composites, 10, 242 (1989). 9. V, Vand, J. Phys. Co&d Chem., 52, 277 (1948). 10. B. Tome, J. Colloid Sci., 4, 511 (1949). 11. Z. B. Jastrzebski, Znd. Eng. Chem. Fun&m., 6, 445 (1967). 12. R. Cox and S. Mason, Ann. Reu. Fluid Me&, 3, 291 (1971). 13. K. K. Trilisskii, G. B. Froishteter, E. L. Smorodinskii, and V.I. Groshchuk, Kolloid Zh., 35, 1109 (1973). 14. G. V. Vinogradov, G. B. Froishteter, K.K. Trilisskii, and E. L. Smorodinskii, Rheol. Acta., 14, 765 (1975). 15. A.M. Kraynik and W. R. Schowalter, J. RheoE., 25, 95 (1981). 16. E. Windhab and W. Gleissle, Proc. IX Z&l. Congress on Rheology, Mexico (1984). 17. Y. Cohen and A. B. Metzner, J. Rheol., 29, 67 (1985). 18. G. Segre and A. Silberberg, Nature, 189, 209 (1961). 19. G. Segre and A. Silberberg, J. Fluid Mech., 14, 115 (1962). 20. V. Seshadri and S.P. Sutera, I ralzs. Sot. Rheol., 14, 351 (1970). 21. C. D. Han, Multiphase Flow in Polymer Processing, Academic Press, New York, U. Yilmazer, SPE ANTEC Technical Papers, 34, 1608 (1988). 23. D. Kalyon, and A. Go&is, SPE ANTEC Technical Papers, 35, 44 (1989). 24. D. Kalyon A. Gotsis, U. Yilmazer, C. Gogos, H. Sangani, B. Aral, and C. Tsenoglou, Adu. Polym. Tech., 8, 4 (1988). 25. E. B. Bagley, J. Appl. Phys., 28, 624 (1957). 26. M. Mooney, J. Rheol., 2, 210 (1931). 27. A. Yoshimura and R. K. Prud homme, J. Rheol., 32, 53 ( B. Rabinowitsch, 2. Physik. Chem., A145, 1 (1929). 29. T. Q. Jiang, A. C. Young, and A. B. Metzner, Rheol. Acta, 25, 397 (1986). 30. W. Kozicki, S. N. Pasari, A. R. K. Rao, and C. Tiu, Chem. Eng. Sci., 25, 41 (1970). Received June 27, 1988 Accepted March 20, 1989