Capillary Extrusion and Swell of a HDPE Melt Exhibiting Slip

Transcription

1 Capillary Extrusion and Swell of a HDPE Melt Exhibiting Slip MAHMOUD ANSARI Department of Chemical and Biological Engineering, The University of British Columbia, Vancouver, BC, V6T 1Z3, Canada EVAN MITSOULIS School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou, , Athens, Greece SAVVAS G. HATZIKIRIAKOS Department of Chemical and Biological Engineering, The University of British Columbia, Vancouver, BC, V6T 1Z3, Canada Received: November 8, 2011 Accepted: April 27, 2012 ABSTRACT: The extrudate (die) swell of a high-density polyethylene (HDPE) melt was studied both experimentally and numerically under slip conditions. The excess pressure drop due to entry (entrance pressure drop), the effect of pressure and temperature on viscosity, and the slip effects on the capillary data analysis have been examined. Using a series of capillary dies having different diameters, D, and length-to-diameter L/D ratios, a full rheological characterization has been carried out and the experimental data have been fitted both with a viscous model (Carreau Yasuda) and a viscoelastic one (the Kaye-Bernstein, Kearsley, Zapas/Papanastasiou, Scriven, Macosko or K-BKZ/PSM model). Particular emphasis has been placed on the effects of wall slip (significant for HDPE). It was found that viscous modeling underestimates the pressures drops (especially at Correspondence to: Savvas G. Hatzikiriakos; savvas. hatzi@ubc.ca. Contract grant sponsor: Natural Sciences and Engineering Research Council (NSERC) of Canada. Contract grant sponsor: National Technical University of Athens (NTUA), Athens, Grrece. Advances in Polymer Technology, Vol. 32, No. S1, E369 E385 (2013) C 2012 Wiley Periodicals, Inc.

2 the higher apparent shear rates and L/D ratios) and predicts virtually no extrudate swell. On the other hand, the viscoelastic simulations were capable of reproducing the experimental data well, and this was particularly true for the pressure drop. The prediction of viscoelastic extrudate swell presented a problem, since the simulations grossly overpredict it due to the highly elastic nature of the melt. This occurs despite the presence of severe slip at the wall, which brings the swell down considerably. At this point it is not clear whether this is due to the viscoelastic model used or other phenomena, such as sagging and/or cooling, when simply extruding in the atmosphere. C 2012 Wiley Periodicals, Inc. Adv Polym Techn 32: E369 E385, 2013; View this article online at wileyonlinelibrary.com. DOI /adv KEY WORDS: Capillary flow, Extrudate swell, HDPE, K-BKZ model, Slip Introduction Capillary rheometry is extensively used in both industry and academia to assess the rheological and processing behavior of polymer melts at high shear rates before testing their processability in full industrial scale. 1 One important aspect of material performance in processing is extrudate (die) swell. 2 4 This is the phenomenon of increasing area or diameter of the extrudate as it comes out from the die, whereupon it suddenly encounters a dramatically different type of flow, i.e., from a constrained flow within the die walls with no-slip or partial slip to a shear-free flow without walls outside the die. As a result of this change in boundary conditions, the polymer swells, sometimes dramatically. 2 The degree of swelling heavily depends on its past deformation history (memory effects), geometric characteristics of the die, and viscoelastic properties. 2 Extrudate swell was designated as a benchmark problem in rheology in the early 1970s. 2 The Newtonian problem was solved first by Tanner, 5 where it was established that Newtonian fluids swell about 13% when exiting from a tube die and 19% when exiting from a slit die, in agreement with experiments. 6 Since then, the majority of efforts have been directed toward the swelling of polymer solutions and melts, where substantial swelling was found experimentally and predicted numerically by a number of viscoelastic constitutive equations, such as the Oldroyd-B model, 7,8 the Phan-Thien/Tanner model, 9,10 and the integral K-BKZ model. 11,12 From the point of view of polymer solutions, success has been quite recently achieved by correctly predicting the extrudate swell of highly elastic Boger fluids with the K-BKZ model. 13 For polymer melts, the first successful simulations of extrudate swell for the highly elastic IUPAC LDPE melt were obtained by Luo and Tanner 11 with the K-BKZ model and verified and extended by Barakos and Mitsoulis 14 and Sun et al.. 15 Meanwhile, the other important polyethylene melt, namely high-density polyethylene (HDPE), was also studied experimentally and computationally 12,20 22 with mixed results. Namely, experiments by Orbey and Dealy 16 from annular dies were simulated with the K-BKZ model by Luo and Mitsoulis 12 and captured the major trends dictated by the die design and the viscoelastic nature of the HDPE melt; other experiments by Park et al. 17 were simulated by Kiriakidis and Mitsoulis 20 but for low apparent shear rates, where the swelling was moderate; and still other experiments by Koopmans 18,19 were simulated by Goublomme et al., 21 and Goublomme and Crochet 22 and showed that various integral models of the K-BKZ type had different degrees of success in predicting the swelling of HDPE, in most cases highly overestimating the experimental values. It is important to note that the situation for HDPE melts is somewhat different from other polymer melts. It is known that HDPE shows significant slip at the wall This greatly alters the deformation history, and if such effects are not taken into account, there is no hope that any constitutive equation would be able to correctly predict the magnitude of extrudate swell. The main objective of the study is to measure and predict the extrudate swell of a HDPE melt that exhibits significant slip effects. It is our goal to demonstrate that such slip effects have to be seriously considered before any reliable predictions E370 Advances in Polymer Technology DOI /adv

3 FIGURE 1. The experimental setup and method used for extrudate swell measurements. Die dimensions (D = 0.79 mm, L /D = 16), T = 190 C, γ A = 100 s 1. can be made on macroscopic quantities related to processing of polymers; in this case extrudate swell is the macroscopic quantity which is significant in processes such as extrusion, wire coating, and blow molding, among others. Experimental MATERIALS A HDPE melt was used in this work carefully selected to address the effects of slip on extrudate swell. This particular HDPE (m-hdpe) has a molecular weight of about 229,800 g/mol and a polydispersity index of Its rheological behavior has been studied previously by Ansari et al. 26 RHEOLOGICAL TESTING As discussed above, the rheology of this resin was studied by Ansari et al. 26 The master curves of its linear viscoelastic moduli are also reported here along with predictions of the K-BKZ constitutive equation for several rheological properties at the reference temperature of 190 C. An Instron capillary rheometer (constant piston speed) was used to determine the extrudate swell and the slip behavior of this polymer. The viscosity as a function of the wall shear stress, σ W,and apparent shear rate, γ A = 32Q/π D 3, where Q is the volumetric flow rate and D is the capillary diameter, is also studied as part of the slip study. Three series of dies having various diameters (D = 0.079, 0.122, and cm) and length-to-diameters ratios (L/D = 5, 16, and 33) were used (in total nine dies) to directly determine the viscosity and the slip behavior through the well-known Mooney analysis at 190 C. 27 The extrudate swell measurements were performed by analyzing extrudate images (immediately after die exit) taken with a high-resolution Nikon D-90 camera equipped with a Sigma DG Macro 2.8 lens attached to three Kenko extension tubes, which give 1.5 magnification in macrofocusing mode. A reference tip with known thickness has been used to estimate the extrudate diameters with respect to its size. Such a typical image is depicted in Fig. 1. The reported extrudate swells for each shear rate are the average of analyzing at least five different images. GOVERNING EQUATIONS AND RHEOLOGICAL MODELING We consider the conservation equations of mass, momentum, and energy for incompressible fluids, under nonisothermal, creeping, and steady flow conditions. These are written as 2,28 : ū = 0 (1) 0 = p + τ (2) ρc p ū T = k 2 T + τ : ū (3) where ρ is the density, ū is the velocity vector, p is the pressure, τ is the extra stress tensor, T is the temperature, C p is the heat capacity, and k is the thermal conductivity. The viscous stresses are given for inelastic non- Newtonian incompressible fluids by the relation 2 : τ = η( γ ) γ (4) Advances in Polymer Technology DOI /adv E371

4 where η( γ ) is the apparent non-newtonian viscosity, which is a function of the magnitude γ of the rate-of-strain tensor γ = ū + ū T, which is given by γ = ( 1 1 ( ) ) 1/2 2 II γ = γ : γ 2 where II γ is the second invariant of γ II γ = ( γ : γ ) = i (5) γ ij γ ij (6) To evaluate the role of viscoelasticity in the prediction of die swell, it is instructive to consider first purely viscous models in the simulations. Namely, the Carreau Yasuda model was used to fit the shear viscosity data of the HDPE melt. The Carreau Yasuda model is written as 1 : j η = η 0 [1 + (λ γ ) α ] n 1 α (7) where η 0 is the zero-shear-rate viscosity, λ is a time constant, n is the power law index, and α is the Yasuda exponent (2 for the simple Carreau model). The fitted viscosity of the HDPE melt by Eq. (7) is plotted in Fig. 2, whereas the parameters of the model are listed in Table I. We observe that the HDPE melt is very shear thinning for shear rates above 1s 1 giving a low power law index n = The Carreau Yasuda model fits the data well over the range of experiment results. It should be noted that our recent paper 29 with another HDPE melt having a higher polydispersity index PDI = 42 (herein desig- Shear viscosity, η (Pa.s) Experimental Carreau Yasuda model Shear Rate, γ (s -1 ) FIGURE 2. The shear viscosity of the HDPE melt at 190 C fitted with the Carreau Yasuda model (Eq. (7)) using the parameters listed in Table I. TABLE I Parameters for the HDPE Melt Obeying the Carreau Yasuda Model (Eq. (7)) at 190 C Parameter Value η 0 191,660 Pa s λ s n α natedashdpe-42)hasusedthecrossmodelforits fitting. However, the Carreau Yasuda model gives a better fit to the present data than the Cross model. Viscoelasticity is included in the present work via an appropriate rheological model for the stresses. This is a K-BKZ equation proposed by Papanastasiou et al. 30 and modified by Luo and Tanner. 11 This is written as τ = 1 t 1 θ N a k exp ( t ) t λ k k=1 α (α 3)+β I C 1 +(1 β)i C [C 1 t (t ) + θc t (t )]dt where t is the current time, λ k and a k are the relaxation times and relaxation modulus coefficients, N is the number of relaxation modes, α and β are material constants, and I C, I 1 C are the first invariants of the Cauchy Green tensor C t and its inverse C 1 t,the Finger strain tensor. The material constant θ is given by λ k N 2 = θ N 1 1 θ (8) (9) where N 1 and N 2 are the first and second normal stress differences, respectively. It is noted that θ is not zero for polymer melts, which possess a nonzero second normal stress difference. Its usual range is between 0.1 and 0.2 in accordance with experimental findings. 1,2 As discussed above, experiments were performed in the parallel plate and extensional rheometers for the HDPE melt to rheologically characterize it. Figure 3 shows plots of the master dynamic moduli G and G for HDPE at the reference temperature of 190 C. The model predictions obtained by fitting the experimental data to Eq. (8) with a spectrum of relaxation times, λ k,and coefficients, a k, determined E372 Advances in Polymer Technology DOI /adv

5 Dynamic moduli, G, G (Pa) Frequency, ω (rad/s) Fit G G FIGURE 3. Experimental data (symbols) and model predictions of storage (G ) and loss (G ) moduli for the HDPE melt at 190 C using the relaxation times listed in Table II. TABLE II Relaxation Spectrum and Material Constants for the HDPE Melt Obeying the K-BKZ Model (Eq. (8)) at 190 C (α = , β = 0.6, θ = 0, λ = 96 s, η 0 = 190,425 Pa s) k λ k (s) a k (Pa) , , , , , , , by a nonlinear regression package, 31 are also plotted. The parameters found from the fitting procedure are listed in Table II. The relaxation spectrum is used to find the average relaxation time, λ, and zeroshear-rate viscosity, η 0, according to the following formulas: λ = N k=1 a kλ 2 k N k=1 a kλ k (10) η 0 = N a k λ k (11) k=1 The values of these parameters are λ = 96 s and η 0 = 190,425 Pa s, indicating an elastic melt with a high average relaxation time. It should be noted that the other HDPE-42 used in our recent paper 29 has Shear (elongational) viscosity, η S(E) (Pa. s) η Ε Ν 1 η S Shear (elongational) rate, γ (ε) (s -1 ) FIGURE 4. Experimental data (solid symbols) and model predictions of shear viscosity, η S, first normal stress difference, N 1, and elongational viscosity, η E,for the HDPE melt at 190 C using the K-BKZ model (Eq. (8)) with the parameters listed in Table II. The N 1 data have been obtained from G and G according to Laun s formula (Eq. (12)). a much lower λ = s and a lower η 0 = 140,073 Pa s. From the data on G and G, it is possible to use Laun s formula to obtain data for the first normal stress difference N 1 according to Dealy and Wissbrun 1 : First normal stress difference, N 1 (Pa) ( G N 1 = 2G [1 ) ] (12) G Figure 4 presents plots of a number of calculated and experimental material functions for the HDPE melt at the reference temperature of 190 C. Namely, data for the shear viscosity, η S, the elongational viscosity, η E, and the first normal stress difference, N 1, are plotted as functions of corresponding rates (shear or extensional). The parameter β that controls the calculated elongational viscosity was fitted by using the extensional behavior of the melt, which is essentially equal to 3η +. It can be seen that the overall rheological representation of all material functions is excellent. NONISOTHERMAL MODELING The nonisothermal modeling follows the one given in earlier publications 11,32 34 and will not Advances in Polymer Technology DOI /adv E373

6 be repeated here. Suffice it to say that it employs the Arrhenius temperature shifting function, a T,given by Dealy and Wissbrun 1 and Tanner 2 : a T (T) = η [ ( E 1 = exp η 0 R g T 1 )] T 0 (13) In the above, η 0 is a zero-shear viscosity at T 0, E is the activation energy constant, R g is the ideal gas constant, and T 0 is a reference temperature (in K). The activation energy constant E can be determined from the shift factors obtained by applying the time temperature superposition to get the master curves plotted in Fig. 2. It was found to be 28,840 J/mol, typical for a HDPE resin. In the present work, we have applied the above equation to derive the nonisothermal constitutive equation from the isothermal one. This method is based on the time temperature superposition principle and simply consists of shifting the relaxation times λ k from the temperature history within the material s internal timescale t. 32 The equation used to shift the relaxation times in the material s history is given by 34 λ k (T (t )) = λ k (T 0 )a T (T (t )) (14) where T is the temperature at time t. The viscoelastic stresses calculated by the nonisothermal version of the above constitutive equation (Eq. (8)) enter in the energy equation (Eq. (13)) as a contribution to the viscous dissipation term. The thermal properties of the melt have been gathered from various sources and are given in our recent publication. 29 The values are reproduced in Table III. TABLE III Values of the Various Parameters for the HDPE Melt at 190 C Parameter Value References β p MPa 1 37,38 β sl 18,800 cm/(s MPa b ) This work b 4.0 This work ρ g/cm 3 54 C p J/(g K) 54 k J/(s cm K) 54 E 28,840 J/mol This work R g J/(mol K) 54 T C (463 K) This work The various thermal and flow parameters are combined to give appropriate dimensionless numbers. 35,36 The relevant ones here are the Peclet number, Pe, and the Nahme Griffith number, Na. These are defined as Pe = ρc pur k Na = ηeu2 kr g T0 2 (15) (16) where η = f (U/R) is a nominal viscosity given by the Carreau Yasuda model (Eq. (7)) at a nominal shear rate of U/R and U(= γ A R/4) is the average velocity in the capillary die. The Pe number represents the ratio of heat convection to conduction, and the Na number represents the ratio of viscous dissipation to conduction and indicates the extent of coupling between the momentum and energy equations. A thorough discussion of these effects in nonisothermal polymer melt flow is given by Winter. 35 With the above properties and a die radius R = 0.04 cm, the dimensionless thermal numbers are in the range: 4 < Pe < 814 and < Na < 1, showing a relatively strong convection (Pe 1) and a weak to moderate coupling between momentum and energy equations (Na 1). A value of Na > 1 indicates temperature nonuniformities generated by viscous dissipation and a strong coupling between momentum and energy equations. More details are given in Table IV. PRESSURE-DEPENDENT MODELING Similarly with the time temperature superposition principle where the stresses are calculated at a different temperature using the shift factor a T,the time pressure superposition principle can be used to account for the pressure effect on the stresses. In both cases of viscous or viscoelastic models, the new stresses are calculated using the pressure-shift factor a p. For viscous models, the following Barus equation is used to modify the viscosity 1 : a p η η p0 = exp(β p p) (17) where η is the viscosity at absolute pressure p, η p0 is the viscosity at ambient pressure, and β p is the pressure coefficient. This coefficient has been reported to be GPa 1 for HDPE. 37,38 E374 Advances in Polymer Technology DOI /adv

7 TABLE IV Range of the Dimensionless Parameters in the Flow of HDPE Melt at 190 C (Die Radius R = 0.04 cm) Apparent Shear Peclet Number, Nahme Number, Pressure-Shift Slip Parameter, Rate, γ A (s 1 ) Pe Na Parameter, B p B sl For viscoelastic models, such as the K-BKZ model (Eq. (8)), the pressure-shift factor modifies the relaxation moduli, a k, according to a k (p(t )) = a k (p 0 )a p (p(t )) (18) This is equivalent to multiplying the stresses by a p, according to Eq. (17). It should be noted that a p is an exponential function of β p, which itself may depend on pressure p, as was the case for the low-density polyethylene (LDPE) melt. 26 Since in a flow field negative pressures may appear, especially around singularities as is the case in contraction flows, special care must be taken numerically to handle these functions for negative numbers. Failure to do so leads to nonsensical results and/or to divergence. The pressure dependence of the viscosity gives rise to the dimensionless pressure-shift parameter, B p. This is defined as B p = β p ηu R (19) When B p = 0, we have no pressure dependence of the viscosity. For the present data, we get < B p < , showing a weak dependence of viscosity on pressure in the range of simulations, unlike the LDPE melt. 39 More details are given in Table IV. SLIP-AT-THE-WALL MODELING In the case of slip effects at the wall, the usual noslip velocity at the solid boundaries is replaced by a slip law of the following form 1,23,24 : u sl = β sl σ b w (20) where u sl is the slip velocity, σ w is the shear stress at the die wall, β sl is the slip coefficient, and b is the slip exponent. As it will be shown below, the values found experimentally for this HDPE melt are β sl = mm/s/mpa b and b = 4. It should be noted that due to its high polydispersity, the selected HDPE does not undergo a stick-slip transition, and therefore both the flow curve and slip velocity are continuous functions. In fact, Eq. (20) describes its complete slip behavior from very small to very high shear rates. For narrow molecular weight HDPEs, and some other melts such as polybutadienes, stickslip may occur; in those cases the slip behavior is a double-valued function of shear stress. 25,40,41 In two-dimensional simulations, the above law means that the tangential velocity on the boundary is given by the slip law, while the normal velocity is set to zero, i.e., β sl ( t n : τ) b = ( t ū), n ū = 0 (21) where n is the unit outward normal vector to a surface, t is the tangential unit vector in the direction of flow, and the rest of symbols are defined above. Implementation of slip in similar flow geometries for a polypropylene (PP) melt has been previously carried out by Mitsoulis et al. 42 and in our recent work Ansari et al. 43 The corresponding dimensionless slip coefficient, B sl, is a measure of fluid slip at the wall: B sl = β sl η b U ( ) U b (22) R When B sl = 0, we have no-slip conditions. When B sl 1, we have macroscopically obvious slip. For the present data, we get < B sl < 1, showing a strong slip effect in the range of simulations, again unlike the LDPE melt, which shows no slip. 39 It should be noted that the other HDPE-42 used in our recent paper 29 has similar slip behavior but a higher slip exponent b = Advances in Polymer Technology DOI /adv E375

8 Method of Solution The solution of the above conservation and constitutive equations is carried out with two codes, one for viscous flows (u-v-p-t-h formulation) 44 and one for viscoelastic flows. 33,45 The boundary conditions (BCs) for the problem at hand are well known and can be found in our earlier publication. 33 Briefly, we assume no-slip (or slip, Eq. (21)) and a constant temperature T 0 at the solid walls; at entry, a fully developed velocity profile v z (r) is imposed, corresponding to the flow rate at hand (found numerically for no-slip or slip conditions), and a constant temperature T 0 is assumed; at the outlet, zero surface traction and zero heat flux q are assumed; on the free surface, no penetration and zero heat flux are imposed. The entry length of the domain is L res = 40R, long enough to guarantee fully developed conditions even for viscoelastic runs for the highest apparent shear rate. The extrudate length depended on whether we used viscous or viscoelastic simulations. For the viscous simulations, there are no memory effects and a relatively short extrudate length L ext = 12R suffices. For the viscoelastic simulations, memory effects are important and longer meshes are necessary. We have tried various extrudate lengths L ext ; however no matter how long the domain was, the calculated swell never leveled off due to the strong viscoelastic nature of the melt. Therefore, we have chosen here to report results for L ext = 16R and this issue will be discussed further in the simulations section. Having fixed the model parameters and the problem geometry, the only parameter left to vary was the apparent shear rate in the die ( γ A = 4Q/π R 3 ). Simulations were performed for the whole range of experimental apparent shear rates, namely from 5s 1 to 1000 s 1, where smooth extrudates were obtained. The viscous simulations are extremely fast and are used as a first step to study the whole range of parameter values. The viscoelastic simulations admittedly are harder to do, and they need good initial flow fields to get solutions at elevated apparent shear rates. In our recent work, 43 we explained how it was possible for the first time to do viscoelastic computations up to very high apparent shear rates (1000 s 1 ) with good results. Here, an extra complication arises from the presence of free surface, for which severe underrelaxation (factor ω f = 0.1 down to 0.01) must be used to avoid particle tracking occurring outside the domain. Briefly, the solution strategy starts from the Newtonian solution at the lowest apparent shear rate (0.1 s 1 ) for the base case (β p = a T = β sl = 0). Then at the given apparent shear rate, the viscoelastic model is turned on and the solution is pursued in the given domain until the norm of the error is below Then the free-surface update is turned on, and the u v p T solution is alternated with the h-solution (free surface location) until the maximum free surface change is less than Meeting this criterion gives a very good solution for the problem at hand. Using this solution as an initial guess, the apparent shear rate is then raised slowly to get a new solution at an elevated value. This way it was possible to achieve solutions for as high as 1000 s 1. It must be noted that HDPE is strongly viscoelastic 12 as is LDPE, 33,46 for which it was not possible to reach apparent shear rates greater than 10 s 1 for the extrudate swell problem (without slip). When all effects are present, we follow the same procedure. Now the biggest contribution comes from slip, since temperature dependence and pressure dependence of the viscosity have opposite effects and they are small anyway. With slip present, the simulations are much faster as they require fewer iterations due to the effectively lower flow conditions encountered in the flow field (actual shear rates at the die walls are an order of magnitude less with slip present). Also the swell is reduced compared with the base case, which makes it easier to solve the nonlinear problem. All velocities have been made dimensionless with the average velocity U and the lengths with the die radius R. Then the pressures and stresses are made dimensionless by η 0 U/R. Experimental Results ENTRANCE (END) PRESSURE Figure 5 presents the apparent flow curves of the HDPE for three dies having the same D and different L/D ratios in terms of the apparent shear stress, σ W,A defined as σ W,A p/(4l/d), versus the apparent shear rate, γ A, where p is the pressure drop along the capillary die including the entry. The data do not superpose due to the fact that the end pressure, p end, has not been taken into account. This E376 Advances in Polymer Technology DOI /adv

9 0.60 Apparent wall shear stress, σ w,a (MPa) L/D = 5 L/D = 16 L/D = Apparent shear rate, γ Α (s 1 ) FIGURE 5. The apparent flow curves of the HDPE melt at 190 C as a function of the apparent shear rate for various L /D ratios. Pressure (MPa) γ 1 ( s A ) L / D FIGURE 6. The pressure drop for the capillary extrusion of the HDPE melt at 190 C as a function of L /D for different values of the apparent shear rate (Bagley plot). can be done by constructing the Bagley plot, which is shown in Fig. 6. The pressure drop for the capillary extrusion is plotted as a function of the die length L/D for an extended range of values of the apparent shear rate from 5 s 1 to 1000 s 1.Thedata fall on straight lines (shown in Fig. 6), indicating that the effect of pressure on viscosity is negligible or that both effects of pressure and viscous heating on viscosity are negligible as these effects point to opposite directions. The values of p end are obtained as points of intersections of the fitted straight lines on the vertical pressure axis. Advances in Polymer Technology DOI /adv E377

10 0.30 Wall shear stress, σ w (MPa) L/D =5 L/D = 16 L/D = Apparent shear rate, γ Α (s 1 ) FIGURE 7. The apparent flow curves of the HDPE melt at 190 C as a function of the apparent shear rate for various L /D ratios corrected for the entrance effects. The data superposes well showing that the pressure effect of viscosity is negligible as expected for HDPE melts. FLOW CURVES AND DIAMETER DEPENDENCE Figure 7 depicts flow curves for the HDPE obtained by using capillaries of different diameter and constant ratio L/D = 16 at 190 C. The diameter dependence of the flow curves is clear. This diameter dependence is consistent with the assumption of slip, and the Mooney technique can be used to determine the slip velocity as a function of shear stress. 27 Also on the same plot, the linear viscoelastic (LVE) data are plotted in the form of a flow curve; in other words, the complex modulus, G G 2 + G 2, is plotted as a function of frequency, ω. The failure of the Cox Merz rule is clear, and this is due to the occurrence of slip, also reported by Ansari et al. 26,43 SLIP-CORRECTED FLOW CURVES AND THE SLIP VELOCITY The data plotted in Fig. 8 can be used to construct the Mooney plot to obtain the slip velocity as a function of the wall shear stress. 27 The Mooney technique is defined by the following relationship: γ A = γ A,s + 8u sl D (23) Wall Shear stress, σ w (MPa) Complex modulus, G * (MPa) LVE D = 1.22 mm D = 2.11 mm Apparent shear rate, γ A (s 1 ) or frequency, ω (rad/s) FIGURE 8. Bagley corrected flow curves of the HDPE melt for different diameters at 190 C. The diameter dependence and the significant deviation from the LVE data (failure of the Cox Merz rule) are consistent with the assumption of slip. where γ A,s is the apparent shear rate corrected for slip effects. Figure 9 is the Mooney plot. The slopes of the straight lines fitted to the data are equal to 8u sl according to Eq. (23). These slopes increase with increasing wall shear stress values. The calculated slip velocity function versus the wall shear stress is plotted in Fig. 10. The values E378 Advances in Polymer Technology DOI /adv

11 Apparent shear rate, γ A (s 1 ) σ w (MPa) / D (mm 1 ) FIGURE 9. Mooney plot using the data plotted in Fig. 8. The slopes of the lines are equal to 8u sl for the corresponding value of stress. The slopes increase with increasing shear stress. Slip velocity, u sl (mm/s) u sl (mm/s)= [σ w (MPa)] 4 Mooney, deviation from LVE D = 1.22 mm, deviation from LVE D = 2.11 mm, deviation from LVE Wall shear stress, σ w (MPa) FIGURE 10. The slip velocity as a function of shear stress for the HDPE melt at 190 C. The solid line represents the slip law given by Eq. (20). log(g )/ log(ω) from the flow curve. All data define a single line indicating consistency of the analysis. Equation (20) was fitted to the data, resulting values of β sl = mm/s/mpa b and b = 4. These numbers indicate a strong nonlinear slip law with a very high exponent b. Based on experimental findings, Funatsu and Kajiwara 47 have reported an exponent of 3.65 for their slip model, Hatzikiriakos and Dealy 24 reported exponents of about 3 3.6, whereas Hill et al. 48 have reported an exponent of 6. Obviously, different HDPE melts slip under different nonlinear slip laws. The corrected capillary flow curve for slip effect alongside with the LVE flow curve are presented in Fig. 11. This figure now shows the validity of the Cox Merz rule for the HDPE, as an excellent superposition is obtained. calculated from the slopes of straight lines are shown as Mooney points. In parallel, slip velocities were calculated from the deviation of each flow curve from the curve indicated as LVE by using the following relationship: where n LVE γ A = 4n LVE 3n LVE + 1 ω + 8u sl D (24) is the local slope defined as n LVE EXTRUDATE SWELL Figures 12a and 12b show the extrudate swell as a function of the apparent shear rate and wall shear stress, respectively. Extrudate swell increases exponentially with an increase in both apparent shear rate and wall shear stress. Moreover, extrudate swell increases with a decrease in the die length, which plays the role of dampening the excitation of the Advances in Polymer Technology DOI /adv E379

12 σ w or G* (MPa) Shifted LVE Mooney D = 1.22 mm D = 2.11 mm γ A (s 1 ) or (4n/3n+1)ω LVE (rad/s) FIGURE 11. The slip corrected flow curve of the HDPE at 190 C compared with the LVE data. Good agreement is shown, demonstrating the validity of the Cox-Merz rule. elasticity effects (normal stresses) at the entry to the capillary. This well-known behavior was first successfully simulated for an LDPE melt (the IUPAC LDPE melt A) by Luo and Tanner. 11 D Extrudate swell Extrudate swell L/D = 5 L/D = 16 L/D = 33 L/D = 5 L/D = 16 L/D = Apparent shear rate, γ Α (s 1 ) (a) Numerical Results VISCOUS MODELING It is instructive to perform first calculations with a purely viscous model, so that the effect of viscoelasticity will become evident later. The numerical simulations have been carried out with the finite element method (FEM) as outlined in the Method of Solution section. For the finite element mesh arrangement, we have used our experience with viscous and viscoelastic flows and chosen a grid that progressively adds more elements as one moves from the reservoir toward the singularity at the entrance to the die, while the elements become larger as one moves away from this entry singularity. Again as the die exit is approached, the elements become smaller due to the exit singularity there, after which the elements progressively become larger. A typical finite element grid is shown in Fig. 13 for L/D = 16 (L/R = 32). The domain represents an 18.75:1 abrupt circular contraction with an entrance angle 2α = 180.Thegrid consists of 1584 elements, 3775 nodes, and 10,260 unknown degrees of freedom (d.o.f.), while a four times denser grid is also used, having been created by subdivision of each element into four subelements for Wall shear stress, σ w (MPa) (b) FIGURE 12. (a) The extrudate swell of the HDPE melt at 190 C as a function of the apparent shear rate for three different L /D values. The extrudate swell decreases with increasing die length. (b) The extrudate swell of the HDPE melt at 190 C as a function of the wall shear stress for three different L /D values. The extrudate swell decreases with increasing die length. checking purposes of grid-independent results. This checking consists of reporting the overall pressures in the system from the two meshes and making sure that the differences are less than 1% between the two results. The viscous numerical simulations have been undertaken with the Carreau Yasuda model (Eq. (7)). This constitutive relation is solved together with the conservation equations of mass, momentum, and energy without or with slip at the wall. Namely, we present two sets of simulations, one called the base case of no effects at all (β p = β sl = a T = 0). Then, all E380 Advances in Polymer Technology DOI /adv

13 FIGURE 13. (a) A typical finite element grid for the simulations in an 18.75:1 abrupt circular contraction with L /R = 32 and 2α = 180. The upper grid (M1) consists of 1584 elements and 5101 nodes, whereas the lower grid is created by subdivision of each M1 element into four subelements to form a denser grid for checking the results for grid-independence; (b) detailed grids near the die entry; and (c) detailed grids near the die exit and extrudate region. effects were turned on, referred to in the graphs as slip (because slip is the dominant effect), so that the differences become evident. Figure 14 presents the pressure drops in the capillary obtained from the simulations (lines) together with the experimental data of Fig. 6 (symbols) for different apparent shear rates and L/D ratios (Bagley plot). The base case simulations (continuous lines) overestimate significantly the experimental data. For example, at γ A = 1000 s 1 and L/D = 33, the experimental pressure drop is P = 26 MPa whereas the simulations result in a pressure drop of P = 39 MPa (an error of +50%). On the other hand, the slip simulations (broken lines) underestimate the experimental data. Again, at γ A = 1000 s 1 and L/D = 33, the simulations give 21 MPa (an error of 19%). Obviously, the inclusion of slip brings the simulation predictions closer to the experimental data, although purely viscous simulations do not predict these well. The situation is even worse when extrudate swell is concerned. The viscous simulations (base case)predict swell ratios that hover around zero swell (from a maximum of +2.8% to 1.6%). The slip simulations never predict negative swell, but they are even closer to zero swell (from +1.3% to 0). These very small swell predictions are well known for purely viscous fluids It is therefore, at this point that we turn our attention to the viscoelastic simulations. VISCOELASTIC MODELING Viscoelastic simulations were performed with the K-BKZ model (Eq. (8)) and the data of Table II. First the simulations did not consider the free surface and the extrudate swell, because the problem is much Advances in Polymer Technology DOI /adv E381

14 Pressure (MPa) γ A ( s ) (slip) 11 (slip) 64 (slip) 390 (slip) 1000 (slip) Carreau Yasuda Model L / D FIGURE 14. The pressure drop for the capillary extrusion of the HDPE melt at 190 C as a function of L /D for different values of the apparent shear rate (Bagley plot). Symbols are experimental data, whereas lines are viscous simulation results with the Carreau Yasuda model (Eq. (7)) and the data of Table I. Solid lines are for the base case (β p = β sl = a T = 0), whereas broken lines are for all effects accounted for (slip). The viscous simulations either overpredict (base case) or underpredict (slip) the experimental data. easier to solve. These simulations provide good results for the pressures because the exit flow does not contribute appreciably to the overall pressure drop in the capillary). 1,52 The results from the simulations are depicted in Fig. 15 with all effects accounted for. Now the viscoelastic predictions are much closer to the experimental data, and in some cases the agreement is excellent (in the case of γ A = 64 s 1 ). If we consider again the data at γ A = 1000 s 1 and L/D = 33, the simulations now give 26 MPa vs. 29 MPa found experimentally (with an error of 10%). If we also consider that there may be a ±10% error in the experimental data, the predictions are very good indeed. Therefore, it is safe to say that the viscoelastic simulations with the K-BKZ model and slip at the wall do a good job in predicting the pressure drops in capillary flow of this highly elastic HDPE melt. The situation for the pressure drop when the exit region and extrudate swell are considered is not affected very much. Typically, the difference in the pressure drop was in the second decimal digit for low-to-moderate shear rates. At higher shear rates, as it will be explained below, it was not possible to obtain reliable solutions with extrudate swell present. The simulations with the exit region present and the accompanying phenomenon of extrudate swell showed very similar results with those reported earlier for HDPE melts Namely, the swell was very high even at low shear rates (below 10 s 1 ) and convergence was lost for shear rates above 100 s 1,because the swell had exceeded 300% up to 16R.Points to be noticed are 1. the melt is very elastic due to high relaxation times and tends to recover the shape it possessed in the reservoir (i.e., to reach 10R and higher in the radial direction); 2. longer dies gave smaller swells, as was also found experimentally, because the material had more distance (and hence time) to relax its stresses; 3. in all our viscoelastic simulations, we have always used irreversibility of the damping function, 45 which was found essential in bringing the swell down by Goublomme et al. 22 ; 4. a nonzero second normal stress difference (θ < 0) reduced the swelling somewhat but not the general trends; and E382 Advances in Polymer Technology DOI /adv

15 Pressure (MPa) γ A ( s ) K-BKZ Model L / D FIGURE 15. The pressure drop for the capillary extrusion of the HDPE melt at 190 C as a function of L /D for different values of the apparent shear rate (Bagley plot). Symbols are the experimental data, whereas lines are simulation results with the K-BKZ model (Eq. (8)), the data of Table II, and all effects accounted for (slip). The viscoelastic model predicts well the pressure drops in the capillary. 5. the simulations are carried out in a way that amounts to isothermal swell, meaning that the material will give out all its viscoelastic character as swelling. This is equivalent to an experiment where the melt is extruded in an isothermal oil bath with the same density and temperature as the exiting melt. 19 The experiments here give nonisothermal swell, as the material is extruded freely and downward in the atmosphere. Therefore, cooling and sagging play an important role, which apparently reduces the swelling appreciably. 19 This is not taking into account by the simulations. The least swell obtained from the simulations was for dies without any reservoir present and with a fully developed velocity profile imposed upstream. These conditions give rise to the asymptotic swell, 2,20 where the material does not have to remember its shape in any reservoir that it originated from (its memory has been fully faded). The results for three dies with L/D = 5, 16, and 33 are given in Fig. 16. In all cases the swell is reported at a distance L ext = 16R. The results depend on the L/D ratio only at the higher range of the shear rates. The swell follows an upward trend similar to the experiments, but the numerical values are much higher than the experimental ones. For example, at γ A = 100 s 1 the numerical swell has reached 50%, whereas the experimental values are 22% (see Fig. 1). At even higher shear rates, the differences become even higher, namely at γ A = 1000 s 1 the calculated swell reaches around 80% 90% whereas the experimental values are as Extrudate swell L/D = 5 L/D = 16 L/D = 33 L/D = 5, θ = K BKZ model Apparent shear rate, γ Α (s 1 ) FIGURE 16. Asymptotic extrudate swell of the HDPE melt at 190 C as a function of the apparent shear rate for three different L /D values. Simulation results with the K-BKZ model (Eq. (8)). Symbols are put to show the continuation steps. The extrudate swell decreases slightly with increasing die length at the higher range of apparent shear rates. Swell values are reported at L ext = 16R. low as 28% for the longest die (L/D = 33). A nonzero second normal stress difference N 2 brings the swell down due to hoop stresses. Thus, for θ = 0.25 (N 2 /N 1 = 0.2 from Eq. (9)), the swell is reduced about 2% to 5%. This is in agreement with previous findings. 10,14,22 Again, it should be emphasized that the experimental values refer to nonisothermal swell, where cooling and sagging are very important and help reduce the swelling considerably. 19 Advances in Polymer Technology DOI /adv E383

16 An attempt was made to add gravity to the flow field forces and extend the extrudate length L ext = 100R. Although this brought down the swell considerably for lower shear rates, for γ A > 10 s 1 the elasticity of the melt made the swell take off again to values similar to those shown in Fig. 16. Perhaps other phenomena play a role in nonisothermal swelling (or nonannealed swelling) of this polymer melt. HDPEs are crystalline polymers, and crystallization may play an important role in the swelling behavior. Upon exiting the die, the outside core of the extrudate crystallizes even at temperatures higher than their equilibrium melting point (flow-induced crystallization) 53 and thus prevents the inside part of the melt from swelling (by applying hoop stresses). This is an important aspect not taken into account in the simulations. Thus here, as in earlier works, 22 the correct prediction of the extrudate swell of HDPE melts remains an elusive subject. Conclusions A HDPE has been studied in entry flows through capillary dies with different L/D ratios (5, 16, and 33) with the purpose of predicting the pressure drop in the system and its extrudate swell. The experiments have shown that this particular HDPE slips strongly at the wall. Full rheological characterization was carried out both with a viscous (Carreau Yasuda) and a viscoelastic (K-BKZ) model. All necessary material properties data were collected for the simulations. The viscous simulations showed that when all effects are taken into account the pressure drops are underpredicted, especially for the higher apparent shear rates and L/D ratios. Also, virtually no extrudate swell is predicted, a well-known deficiency of viscous models. The viscoelastic simulations with the K-BKZ/ PSM model showed a good predictive capability of the pressure drops in the system for all cases. However, the swell was overpredicted, despite the fact that slip was included. The general trend of the experimental data that showed an exponential increase of extrudate swell was captured by the model, albeit for the asymptotic swell that does not take into account the presence of the reservoir. An increase in the L/D ratio reduces the swelling, and this was more evident experimentally as a long die gives enough time to the material to forget its elastic stresses (fading memory). References 1. Dealy, J. M.; Wissbrun, K. F. Melt Rheology and Its Role in Plastics Processing Theory and Applications; Van Nostrand Reinhold: New York, Tanner, R. I. Engineering Rheology, 2nd ed.; Oxford University Press: Oxford, UK, Boger, D. V.; Walters, K. Rheological Phenomena in Focus, Rheology Series, Vol. 4.; Elsevier, Amsterdam, Bird, R. B.; Hassager, O.; Armstrong, R. C.; Curtiss, C. F. Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory; 2nd ed.: Wiley: New York, Tanner, R. I. Appl Polym Symp 1973, 20, Middleman, S.; Gavis, J. Phys Fluids 1961, 4, Crochet, M. J.; Keunings, R. J Non-Newtonian Fluid Mech 1982, 10, Crochet, M. J.; Keunings, R. J Non-Newtonian Fluid Mech 1982, 10, Bush, M. B.; Tanner, R. I.; Phan-Thien, N. J Non-Newtonian Fluid Mech 1985, 18, Sugeng, F.; Phan-Thien, N.; Tanner, R. I. J Rheol 1987, 31, Luo, X.-L.; Tanner, R. I. Int J Num Meth Eng 1988, 25, Luo, X.-L.; Mitsoulis, E. J Rheol 1989, 33, Mitsoulis, E. J Non-Newtonian Fluid Mech 2010, 165, Barakos, G.; Mitsoulis, E. J Rheol 1995, 39, Sun, J.; Phan-Thien, N.; Tanner, R. I. Rheol Acta 1996, 35, Orbey, N.; Dealy, J. M. Polym Eng Sci 1984, 24, Park, H. J.; Kiriakidis, D. G.; Mitsoulis, E.; Lee, K.-J. J Rheol 1992, 36, Koopmans, R. J. Polym Eng Sci 1992, 32, Koopmans, R. J. Polym Eng Sci 1992, 32, Kiriakidis, D. G.; Mitsoulis, E. Adv Polym Technol 1993, 12, Goublomme, A.; Draily, B.; Crochet, M. J. J Non-Newtonian Fluid Mech 1992, 44, Goublomme, A.; Crochet, M. J. J Non-Newtonian Fluid Mech 1993, 47, Hatzikiriakos, S. G.; Dealy, J. M. J Rheol 1991, 35, Hatzikiriakos, S. G.; Dealy, J. M. J Rheol 1992, 36, Hatzikiriakos, S. G.; Dealy, J. M. J Rheol 1992, 36, Ansari, M.; Hatzikiriakos, S. G.; Sukhadia, A. M.; Rohlfing, D. C. Rheol Acta 2011, 50, Mooney, M. J Rheol 1931, 2, Mitsoulis, E.; Hatzikiriakos, S. G. J Non-Newtonian Fluid Mech 2009, 157, Ansari, M.; Hatzikiriakos, S. G.; Mitsoulis, E. J Non- Newtonian Fluid Mech 2012, , Papanastasiou, A. C.; Scriven, L. E.; Macosko, C. W. J Rheol 1983, 27, Kajiwara, T.; Barakos, G.; Mitsoulis, E. Int J Polym Anal Character 1995, 1, Alaie, S. M.; Papanastasiou, T. C. Intern Polym Proc 1993, 8, E384 Advances in Polymer Technology DOI /adv

17 33. Barakos, G.; Mitsoulis, E. J Non-Newtonian Fluid Mech 1996, 62, Beaulne, M.; Mitsoulis, E. J Appl Polym Sci 2007, 105, Winter, H. H. Adv Heat Transfer 1977, 13, Mitsoulis, E.; Wagner, R.; Heng, F. L. Polym Eng Sci 1988, 28, Sedlacek, T.; Zatloukal, M.; Filip, P.; Boltizar, A.; Saha, P. Polym Eng Sci 2004, 44, Carreras,E.S.;ElKissi,N.;Piau,J.M.;Toussaint,F.;Nigen, S. Rheol Acta 2006, 45, Ansari, M.; Zisis, Th.; Hatzikiriakos, S. G.; Mitsoulis, E. Polym Eng Sci, 2012, 52, Lupton, L. M.; Regester, J. W. Polym Eng Sci 1965, 5, Park, H. E.; Lim, S. T.; Smillo, F.; Dealy, J. M. J Rheol 2008, 52, Mitsoulis, E.; Kazatchkov, I. B.; Hatzikiriakos, S. G. Rheol Acta 2005, 44, Ansari, M.; Alabbas, A.; Mitsoulis, E.; Hatzikiriakos, S. G. Int Polym Proc 2010, 25, Hannachi, A.; Mitsoulis, E. Adv Polym Technol 1993, 12, Luo, X.-L.; Mitsoulis, E. Int J Num Meth Fluids 1990, 11, Luo, X.-L.; Mitsoulis, E. Int J Num Meth Fluids 1990, 11, Funatsu, K.; Kajiwara, T. In Encyclopedia of Fluid Mechanics, Vol. 7: Rheology and Non-Newtonian Flows; Ed. Cheremisinoff, N. P., Gulf Publishing Company, Houston, TX, 1988, pp Hill, D. A.; Hasegawa, T.; Denn, M. M. J Rheol 1990, 34, Mitsoulis, E.; Vlachopoulos, J.; Mirza, F. A. Polym Eng Sci 1984, 24, Mitsoulis, E. J Fluids Eng 2007, 129, Mitsoulis, E. J Non-Newtonian Fluid Mech 2007, 141, Mitsoulis, E.; Vlachopoulos, J. J Polym Eng 1985, 5, Sentmanat, M.; Delgadillo, O.; Hatzikiriakos, S. G. Rheol Acta 2010, 49, Tadmor, Z.; Gogos, C. G. Principles of Polymer Processing, SPE Monograph Series; Wiley: New York, Advances in Polymer Technology DOI /adv E385