I
A. B. METZNER University of Delaware, Newark, Del.
Fracture of Non-Newtonian Fluids at High Shear Stresses
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Extrusion rates of molten polymers may be limited by the stresses at which irregularities in flow rates and product shape begin. The irregularities increase rapidly with further increases in stress and may lead to fracture of the product filament, tube, or sheet. This article reviews the prior art, to help define the causes of this phenomenon and means for its control
SEVERAL
recent reports are concerned with the fracture of viscous fluids. To elucidate the basic mechanism new approaches are desirable; this report attempts to separate valid and invalid conclusions and to suggest directions for further study. The basic problem concerns the mechanics of fluid jets. When fluids flow downward through a veritical tube, two distinct types of behavior may occur i n emerging jets. For low viscosity fluids such as water, which are believed to be Newtonian in behavior, jet diameter gradually decreases with increasing distance from the end of the tube. This is caused partly by progressive increases in the velocity of the fluid resulting from gravitational forces, and partly by changes in velocity profile. If a fluid is in turbulent flow within the tube, its velocity profile is nearly the same as that of thejet; ifin laminarflow, its profile has to change from parabolic at the exit of the tube to perfectly flat further downstream. By force and material balances on the jet, it can be shown that this change in velocity profile alone doubles the average fluid velocity and therefore causes a 40% radial contraction of the jet. On the other hand, jet behavior of a high viscosity oil or polymer melt is different, even if the material appears to be Newtonian in shear. Instead of gradually decreasing in diameter, the jet bulges as soon as it leaves the tube. Further downstream, it again contracts under the influence of gravity and velocity redistribution. As the mean velocity increases, the initial bulge also increases. With polymer melts, jet diameters 2.5 to 2.8 times greater at the tube diameter have been reported; two-
fold increases in diameter are easy to obtain. For polymer melts which are gonNewtonian in nature, another phenomenon has been observed. As velocity increases, the jet of fluid begins to oscillate with progressive violence until it fractures (25, 26). All published quantitative reports deal only with the flow of polymer melts. However, a similar phenomenon has long been known qualitatively to investigators of dilatancy, and these observations have been extended by Riggs (23) and by Whitlock (73, 30). Experimental Facts Westover and Maxwell (29) have determined shear stresses and the corresponding values of 4q/rR3 at which roughness first occurs in polyethylene melts extruded through short capillaries ( L / D= 20) of various diameters, at temperatures between 120' and 204' C. I n the larger tubes, this "critical" value of shear stress was independent of temperature; although it decreased slightly with increasing temperatures in smaller tubes this variation was perhaps no greater than the error involved in attempting to define the point at which extrudate irregularities first appear. Increasing the tube diameter over the sixfold range studied progressively decreased the shear stress corresponding to the initial occurrence of irregularities. Studies by Tordella (26, 28) and Spencer and Dillon (25) on fracture of a variety of polymers showed that this critical shearing stress corresponding to incipient occurrence of irregularities was temperature-independent for polyethyl-
ene from 130' to 240°, for polystyrene from 210' to 260°, and for methacrylate from 140' to 220' C. For polytetrafluoroethylene, no precise data are available from which the critical shear stress may be calculated as a function of temperature. Only shear rates were published; energies of activation to convert these values to shear stresses are available ( 7 7) for all these polymers except polytetrafluoroethylene to at least two significant figures. However, shear stresses calculated using an estimated activation energy of 15 f 5 kcal. per gram mole show no variation with temperature within this uncertainty. For nylon, the critical shearing stress decreased rapidly with increases in temperature, but data are available only over a smaller (50' C.) temperature range. Correction of Tordella's (26) polyethylene data for end effects by the method of Bagley ( 2 ) indicates no variation in the critical shearing stress with tube diameter over a 2.5-fold range. Severs' (24)data show the same trends as Tordella's: Threefold variations in tube diameter caused no change in shear stress corresponding to the intial irregularities in cellulose acetate and poly(methyl methacrylate). This stress was independent of temperature within the experimental accuracy of 5 2 5 % for polyethylene between 160' and 225' C., polystyrene between 175' and 225', and methacrylate between 175" and 200' C. Moisture had a pronounced effect on the critical shearing stress of cellulose acetate; hence reproducible drying of the polymer was necessary. Bagley (2) showed that entrance pressure losses in capillary tubes were a VOL. 50, NO. 10
OCTOBER 1958
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strong function of shear rate; dependence on shear rate changed rapidly as conditionscorresponding to extrudate roughness were approached. Spencer and Dillon (25) also determined the effect of polymer molecular weight. With increasing molecular weight critical stress decreased; over the entire range of molecular weights studied (196,000 to 527,000) the product of molecular weight and critical shearing stress was a constant. Riggs (23) found that dilatant starch suspensions not only formed a rough or vibrating extrudate but tore and finally ruptured sufficiently to produce a fine spray, as the shearing stress was progressively increased. However, the entrance region pressure losses were too high for precise definition of the shearing stress a t which rupture occurred. Whitlock (73, 30) extended these studies by shearing titanium dioxide suspensions in both a rotational (Stormer) viscometer and a capillary tube: Fracture was not only visible at sufficiently high stresses, but also always plainly audible. The critical shearing stress was identical in both instruments. Conclusions. The critical shearing stress corresponding to incipient irregularities in polymer extrudates is independent of both temperature and diameter of the capillary used. The effect of temperature has been defined by study of six different polymer types; with the sole exceptions of the data on 66 nylon (studied only over a modest temperature range) and polyethylene (23), all data are consistent. I t is possible that temperature-dependent effects occurred in the nylon used. As polyethylene was also studied by two other investigators (24, 27) and no temperature effect was noted, it appears that the data of Westover and Maxwell are in error, possibly because of invalid pressure loss corrections. [Westover and Maxwell attemped to correct shear stresses for entrance losses by measuring the pressure drop through a sharp-edged orifice of the same diameter as the tube, and correcting the tube pressure drops by the same amount. However, not all of the eetrance effect is confined to this region-a significant fraction of the excess pressure loss due to flow into a contracted tube occurs in the tube itself (6, 7,9) and is not accounted for in the corrections used by these authors.] Similarly, the peculiar effect of tube diameter found by Westover and Maxwell appears to be in error; neither the data of Spencer and Dillon (25) nor those of Severs (24) show similar effects, even over the same range of tube diameters. The capillary tubes of Spencer and Dillon were of the same diameters as some used by Westover and Maxwell. Tordella has recently extended proof of the absence of a tube diameter effect to a 30-fold range of tube diameters (27).
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irregularities in the work of Westover Tordella and Maxwell is 1.6 X (28) has shown that for other polymers these numbers are also many orders of magnitude lower than those which would correspond to the onset of turbulence. Thus, extrudate irregulariries must be due to other causes. Tordella (26, 28) concluded that extrudate irregularities were due to flow irregularities in the region just upstream from the capillary tube entrance. In a beautifully documented paper (28) he showed that such flow irregularities near the entrance to the tube indeed exist. Theoretical interpretations These effects are further documented by Bagley’s (2) study of entrance pressure losses. However, the differentiation beNason (74)and Westover and Maxwell tween cause and effect is not enrirely (29) believed extrudate irregularities to clear; Corrsin (4) has pointed out that be due to turbulence; however (6, 70, flow irregularities in the capillary itself, 72), the many literature references to or even at the exit, could be responsible “structural” turbulence rest chiefly upon for the observed irregularities in the data taken by Ostwald, Rabinerson, and material chamber and tube entrance, d a C. Andrade (7, 5, 76, 77, 20). rather than vice versa. Unfortunately, These are a t best of questionable validno experimental data have been pubity-da C. Andrade and Lewis studied lished on floib through extremely long secondary flow effects but not turbulence capillaries, in which such an upsrream quantitatively, and Ostwald and Rabinpropagation was difficult or impossible erson were frequently unable to achieve and which would enable the necessary true turbulence even with water. The differentiation. Some supporting evismall inflections in the shear stress-shear dence for Corrsin’s position is found in the rate curves for water frequently found by work of Whitlock (3@. A long capillary these investigators are an order of magtube, in which the approximate location nitude less pronounced than turbulence of the audible fracturing noise could be would be, and must therefore be ascribed determined, was used. This fracture solely to experimental errors, end correcappeared to be located near the exit tions, etc. Several fluids studied by rather than inlet of the tube. However, Ostwald and coworkers have since been the mechanism of rupture in dilatant reported to possess dilatant propertiesslurries may be different from that in phenomena which could easily be mispolymer melts. taken with evidence of turbulence. It is Spencer and Dillon (25) and Severs evident that “structural” turbulence has (24)also assumed the exit of the capillary, little if any basis in fact at present. rather than its inlet, to be the location On the basis of extensive experiof the causes of fluid fracture. They mental data (6, 7, 72) it has been shown theorized that progressive increases in that turbulence in non-Sewtonian sysmolecular orientation must accompany tems does not occur until generalized Reynolds numbers (N’R~ =Dn’V2-”’p/y) increase in shearing rate as one moves radially from the center line of the tube of the order of 2100 are reached, as in the toward the wall. The re-randomizing case of Newtonian behavior. As the of molecular orientation as shearing flow behavior index, n‘, of the fluid destresses (hence shearing rates) are recreases-Le., as the fluid becomes promoved upon emergence of the fluid from gressively more pseudoplastic-the genthe tube would cause a greater contraceralized Reynolds number corresponding tion a t the surface of the filament than at to the onset of turbulence increases somethe center line; hence the observed what, reaching a value of about 3100 for buckling, or at sufficiently high rates of n’ = 0.4 (6, 7). Preliminary results (6, contraction, fluid fractures. 7) indicate a very strong additional supThe theories of Tordella and of pression of turbulence in systems which Spencer, Dillon, and Severs are qualitaare highly viscoelastic, as most polymer tively in agreement with several experimelts are, Thus, all available data supmental facts: port the conclusion that flow will be laminar if the generalized Reynolds As temperature increases, the acnumber is of such a magnitude that lamcompanying decrease in apparent visinar flow would prevail if the system cosity should permit higher rates of conwere Newtonian. Generally Reynolds traction in the Spencer-Dillon mechnumbers of 2100 will be required and anism, and aid uniform flow in the under no conditions would turbulence inlet region to a tube in the Tordella occur in a round tube at generalized mechanism. This is in agreement with Reynolds numbers below about 1000. the experimental observation that the The maximum value of the generalized shear stress at incipient buckling is independent of temperature, permitting inReynolds number at the inception of
The suspensions used by Whitlock (30) and Riggs (23) appear to behave like polymer melts. The effect is independent of the type of viscometric equipment used in this case. Shearing stresses of the order of magnitude of 106 to 108dynes per sq. cm. (2 to 200 atm.) lead to rupture of polymer melts with internal pressures of several thousands of atmospheres. I n the dilatant aqueous suspensions, rupture sometimes took place a t stresses below 500 dynes per sq. cm. (5 X lop4atm.).
INDUSTRIAL AND ENGINEERING CHEMISTRY
VISCOUS FLUID FRACTURE creases in shear rate directly proportional to the changes in apparent viscosity with temperature. Fracture or irregularity in polymer extrudates has apparently been observed only at shear stresses falling in the nonNewtonian (pseudoplastic) region of a shear stress-shear rate curve. This is in agreement with the theory that in the Newtonian region [in which little or no molecular alignment takes place (3, 77)] no forces tending to cause either flow irregularities or buckling would be manifested appreciably. These considerations suggest that irregularities and fracture observed in dilatant suspensions are due to a different mechanism than in polymer melts. I n suspensions of nearly equidimensional particles no alignment would be expected ; however, the volumetric dilation at increasing shear stresses would be accompanied by a flow mechanism in which the “slip” of adjacent layers gave rise to the velocity gradient, rather than the movement of individual molecules, particles, or groups of particles from one position in the fluid to another. Thus in the case of dilatant suspensions the fracture would be more nearly like the “slip” observed in metal crystals than the buckling of a filament or oscillation of flows in the inlet region of a tube. Spencer and Dillon (25) have stated that in polystyrene the ratio of extrudate to capillary tube diameter at incipient irregularity is independent of the tube dimensions, polymer molecular weight, and temperature over considerable ranges of each variable. As shear moduli are nearly temperature-independent, over the range considered, this suggested that the elastic (recoverable) strain was independent of temperature. Severs’ (24) data on polystyrene do not check those of Spencer and Dillon. Severs found n[n =(a’/Q2] to change from 6.0 to 6.5 at 175’ C. to 7.5 to 8.0 a t 225” C., compared to 3.0 =t0.2 found by Spencer and Dillon for temperatures between 175’ and 250’ C. However, in the case of polyethylene Severs found tfi to be constant a t about 3.8 over a n 85’ C. temperature range. Severs’ tubes were much shorter than those of Spencer and Dillon, however. As it is not clear whethei this or experimental errors lead to the incompatible conclusions of these two groups of investigators, discussion of this evidence serves to weaken, but not reject, the Spencer-Dillon hypothesis. Reiner (27) has developed equations for predicting failure in simple materials such as Kelvin and Maxwell bodies. The fact that polymer melts cannot be characterized by such simple models (78) precludes quantitative application of this development. Similarly, Noll’s (75) prediction of fracture in a threedimensional Maxwell body cannot be
accepted, as the same development does not predict shear stress-shear rate curves in agreement with experimental facts. Further, Noll’s approach predicts no breakdown of dilatant fluids, whereas such breakdown may occur at shear stresses very close to those a t which dilatancy is first observed. Recent studies (3, 79) have shown that normal stresses in viscoelastic systems may be larger than corresponding shearing stresses. As these develop longitudinal tensile stresses in the fluid flowing through a capillary (79), their presence may lead to rupture. Many experimental facts are in agreement with such a supposition. This hypothesis would predict that irregularities first occur a t the same shearing stress in all types of viscometers, as the magnitude of tlie stress rather than its distribution (as in the Spencer-DilIon and Severs’ theories) is of importance. While irregularities of the type of interest here have apparently also been observed, in a rotational viscometer ( 8 ) , no quantitative data on their onset were obtained. The last three theories predict three different locations for the effects leading to the phenomenon of extrudate irregularity: Tordella concludes that the important effects occur a t the tube entrance; Spencer, Dillon, and Severs conclude that effects a t the exit are responsible; the normal stresses hypothesis predicts effects within the tube itself or at the inlet. Available data on polymers do not enable the defiqite choice of one of these. The data on fracture of dilatant suspensions could be explained only by the normal stresses theory, assuming such systems were sufficiently elastic or possessed a high coefficient of “cross viscosity” (22). However, a fourth mechanism-slip between layers of particles-may account for the fracture of dilatant suspensions and possibly also polymer fracture. Future investigators of fracture in polymers might do well to vary the type of viscometric equipment used, as well as its dimensions, to aid in selecting the correct fracture mechanism. Acknowledgment
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The comments made by J. P. Tordella on revising the manuscript are acknowledged with thanks. Nomenclature
Any consistent set of units may be used.
d
D g,
K’ L n’
= diameter of extrudate = inner diameter of capillary tube = dimensional conversion factor 3 fluid consistency index , = length of capillary tube = flow behavior index
NRe = generalized Reynolds number, D”’T/’z-n’p N ’ R= ~ I
= =
?!
V
=
E
=
p y
=
=
volumetric flow rate radius (inner) of capillary tube linear velocity, volumetric average change in cross-sectional area of extrudate, = (d/D)z fluiddensity denominator of generalized Reynolds number, y =
g,K’8“ ’- 1
literature Cited
(1) Auerbach, R., Feldman, J., Ostwald,. Wo., Kolloid Z . 43, 155 (1927). (2) Bagley, E. B., J . Appl. Phys. 28, 624 (1957). (3) Brodnyan, J. G., Gaskins, F. H., Philippoff, W., Trans. SOC.Rheol. 1, 109 (1957). (4) Corrsin, S., private communication, 1956. (5) da C. Andrade, E. N., Lewis, J. W., Kolloid Z . 38, 260 (1926). (6) Dodge, D. W., Ph.D. thesis, Univ. of Delaware, Newark, 1957. (7) Dodge, D. W., Metzner, A. B., A.I. Ch.E. meeting, Chicago 1957. (8) Fensom, D. S., Can. J . Technol. 33, 194 (1955). (9) McMillen, E. L., Chem. Engr. Progr. 44, 537 (1948). (10) Metzner, A. B., “Advances in Chemical Engineering,” T. B. Drew and J. W. Hoopes, eds., vol. 1, pp. 77-153, Academic Press, New York, 1956. (11) Metzner, A. B., in “Thermoplastics Processing,” E. C. Bernhardt, ed., Reinhokd, New York, in press. (12) Metzner, A. B., Reed, 3. C., A.Z.Ch.E. Journal 1, 434 (1955). (13) Metzner, A. B., Whitlock, M., Society of Rheology, 1957. (14) Nason, H. K., J . Appl. Phys. 16, 338 (1945) (15’) Noll, W., J . Ratiqnal Mech. and Analysis 4, 3 (1955). (16) Ostwald, Wo., Kolloid 2. 68, 211 (1934). (17) Ostwald, Wo., Auerbach, R., Zbid., 38, 261 (1926). (18) Pao, Y., J . Appl. Phys. 28, 591 (1957). (19) Philippoff, W., Trans. Sac. Rheol. 1, 95 (1957). (20) Rabinerson, A., Fuchs, G., Kollotd Z . 65, 307 (1933). (21) Reiner, M., “Deformation and Flow,” H. K. Lewis, London, 1949. (22) Reiner, M., “Rheology,” F. R. Eirich, ed., vol. I, pp. 9-62, Academic Press, Ne4 York, 1956. (23) Riggs, L. C., B.Ch.E. thesis, Univ. of Delaware, Newark, 1956. (24) Severs, E. T., Ph.D. thesis in chemical engineering, Univ. of Delaware, Newark, 1950. (25) Spencer, R. S., Dillon, R. E., J . Colloid Sci. ’, 241 (1949). (26) Tordella, J. P., J . Appl. Phys. 27, 254 (1956). (27) Tordella, J. P., SPE Journal 13, No. 8, 36 (1957). (28) Tordella, J. P., Trans. SOC.Rheol. 1,203 I
I1 957) ,-~-.,.
(29) Westover, R. F., Maxwell, B., SPE Journal 13, No. 8 , 27 (1957). (30) Whitlock. M.. M.Ch.E. thesis, Univ. . of Delaware; Newark, 1957. RECEIVED for review October 9, 1957 ACCEPTED July 17, 1958 +Or. 50, NO. 10
OCTOBER 1958
1579
Vulcaniza ble Saturated Acrylate Elastomers
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Correction
X
M O L E S OF FORMALDEHYDE
M O L E S OF FORMALDEHYDE
In the article on “Vulcanizable Saturated Acrylate Elastomers” [Fred Leonard, Joshua Nelson, George Brandes, IND.ENG. CHEM.50, 1053 (1958)l an error was made in printing Figure 9 (page 1057). This figure should have appeared as three graphs, as shown here.
Figure 9. Number of cycles to break increases, and swelling index and solubility decrease with increase in formaldehyde concentration up to 2 moles of formaldehyde
0
1 580
1/2 I 2 3 MOLES OF FORMALDEHYDE PER MOLE OF METHACRYLAMIDE
INDUSTRIAL AND ENGINEERING CHEMISTRY
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