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Quantification of the slip phenomenon and the effect of shear thinning in the capillary flow of linear polyethylenes.

INTRODUCTION

An outstanding issue in rheology is the "flow en-Ahancement" that some polymeric liquids present. Flow enhancement has been associated with the violation of the zero-velocity condition at an immobile wall. This phenomenon, also known as "slip," has been documented for a long time, starting with some of the pioneers of rheology (1, 2). and has been recently reviewed by Denn (3). The understanding and proper evaluation of the slip phenomenon may lead to new procedures for rheological characterization and optimization in transformation processes of polymers.

Different types of flow enhancement have been recognized, which can be roughly divided into "true" and apparent" slip, whether the no-slip boundary condition fails or not. This work will be concerned only with slip in polymer melts in Poiseuille type flows. A description of the different types of slip occurring in polymer melts is given by El Kissi and Piau (4).

The slip phenomenon has been in most cases quantified by a slip velocity ([V.sub.s]) introduced by Mooney (2), which is a phenomenological correction representing the contribution of slip to the whole average fluid velocity. More recently, the slip length or extrapolation length "b" was introduced by Brochard and de Gennes (5) and has been extensively used by Wang (6) to characterize the slip behavior.

The slip phenomenon, as it occurs in shear flow, has been found to depend on processing variables as well as on parameters related to the polymer architecture, like molecular weight, polydispersity, and degree of branching (6, 7). Thus, in the presence of slip, the whole flow behavior is determined by a competition between slip and shear flow, this last producing shear thinning in most polymer melts.

The effect of shear thinning in a flow with slip has been considered by several authors. Hatzikiriakos and Dealy (7) proposed that shear thinning could produce a decrease in the slip velocity for increasing shear stress. Yang et al. (8) and Mhetar and Archer (9) observed a decrease in the extrapolation length with increasing stress, which was ascribed to shear thinning. Recently. Joshi et al. (10) also predicted the same behavior for "b" through their unified transient network model for slipping fluids, which was attributed to chain stretching or disentanglement. In most cases, the slip velocity in polymer melts has been reported as an increasing function of the shear stress, so reports on decrease of the slip velocity could be no more than a decrease in the contribution of slip to the whole average fluid velocity.

On the other hand, the physicochemical interaction between polymer melts and die materials has been found to be relevant in polymer processing. The interplay between polyethylene melts and capillary dies constructed from different metals was explored long ago by Benbow and Lamb (11) with respect to the onset of melt fracture in low-density polyethylene (LDPE). These authors found that melt fracture occurred at higher shear stresses when brass dies are used, as compared to those observed in steel, because of reduced slip. However, Metzger and Hamilton (12) reported independence of the extrusion instabilities on die materials for high-density polyethylene (HDPE). Later, Ramamurthy (13) reported the continuous processing of linear low-density polyethylene (LLDPE) without extrudate distortions with brass dies, while distortions were observed during the extrusion through a stainless steel die. Ramamurthy attributed his results to enhanced adhesion between the polymer and the wall. Ramamurthy's results were recentl y reproduced by Ghanta et al. (14) and Perez-Gonzalez and Denn (15), but in contrast to Ramamurthy, the elimination of extrudate distortions was attributed by these authors to flow enhancement or slip, which was, in some cases, strong enough to produce electrification of the melt (16).

In the present work, we perform a simple macroscopic analysis that allows describing the trend of the slip length and the contribution of the slip velocity to the whole average fluid velocity as functions of the shear stress. In addition, the dependence of the slip velocity and the extrapolation length on the polymer architecture and their relation to the shear-thinning behavior are qualitatively described. Thus, it is shown that an increase or decrease of the slip length as the shear stress is increased is completely determined by the polymer characteristics, and by the ratio between the slip and average fluid velocities. The analysis is validated using experimental data obtained during the continuous extrusion of LLDPE melts under strong slip conditions in a brass die.

THEORY

The rheometric approach to slip quantification in capillary flow relies on the well-known Mooney (2) equation, in which the slip velocity ([V.sub.s]) in a capillary of diameter D is related to the apparent wall shear rate ([[gamma].sub.w]) by:

[[gamma].sub.w] = 8 [V.sub.s]/D + 4/[[tau].sup.3.sub.w][[integral].sup.[[tau].sub.w].sub.0][gamma]([tau ])[[tau].sup.2]d[tau] (1)

in which, [[tau].sub.w] is the wall shear stress and [gamma]([tau]) is the constitutive relation for the shear rate. [[gamma].sub.w] is given by 32[Q.sub.exp]/[pi][D.sup.3] where [Q.sub.exp] is the experimentally measured flow rate. An equivalent to Eq 1 in terms of volumetric flow rates is:

[Q.sub.exp] = [Q.sub.s] + [Q.sub.o] (2)

where [Q.sub.s] and [Q.sub.o] represent the contribution to the flow rate due to slip and shear flow, respectively.

Since the shear flow behavior should not depend on the rheometer dimensions, Eq 1 shows that for a given wall shear stress, [[tau].sub.w], the slip velocity may be determined from the slope of the [[gamma].sub.w] versus 1/D.

Thus, once the slip velocity has been determined, its contribution to the whole average fluid velocity (<V>) can be quantified as the ratio:

[V.sub.s]/<V> (3)

where <V> = 4[Q.sub.exp]/[pi][D.sup.2]. The largest possible value

that the ratio [V.sub.s]/<V> can take, one, corresponds to a

flat velocity profile in the capillary, occurring under pure slip conditions, while the zero value is attained for a slip-free flow. Note, however, that the value of one is not physically possible, since for a fluid with no yield stress, a shear flow component is always present.

The slip length "b" is usually calculated as suggested by Wang (6):

[Q.sub.exp] - [Q.sub.o]/[Q.sub.o] = 8b/D (4)

Note that a pure slip flow ([Q.sub.o] = 0) leads to a physically inconsistent result, i.e., to an infinite "b" value. The left-hand side of Eq 4 represents the ratio between the slip velocity ([V.sub.s]) and the slip-free average velocity (<[V.sub.o]>) as obtained from Eq 2, then we can write:

b [varies] [Q.sub.exp] - [Q.sub.o]/[Q.sub.o] = [V.sub.s]/<[V.sub.o]> (5)

In addition, let us assume a power law behavior for the slip velocity and shear flow, i.e. [V.sub.s] = [K.sub.1] [[tau].sub.w.sup.p] while <[V.sub.o]> = [K.sub.2] [[tau].sub.w.sup.1/n], respectively, "n" being the shear thinning index, [K.sub.1] and [K.sub.2] constants for a given temperature and "p" dependent on the polymer characteristics. Thus we can rewrite Eq 5 as:

b [varies] [K.sub.1] [[tau].sup.p.sub.w]/[K.sub.2] [[tau].sup.1/n.sub.w] = [K.sub.1]/[K.sub.2] [[tau].sup.p-(1/n).sub.w] (6)

Then for a given temperature and a polymer-die pair:

b [varies] [[tau].sup.p-(1/n).sub.w] (7)

From Eq 7, a log-log plot of "b" or [V.sub.s]/<[V.sub.o]> versus [[tau].sub.w] would render a straight line with a slope that will be determined by the "p - 1/n" value. If p > 1/n, then p - 1/n > 0, and therefore slip predominates over shear thinning, and "b" or the ratio [V.sub.s]/<[V.sub.o]> should increase along with the shear stress. Otherwise, p < 1/n implies p - 1/n < 0, and then shear-thinning predominates over slip and both "b" and [V.sub.s]/<[V.sub.o]> should decrease as the shear stress is increased. Thus, for a given polymer-die pair, the dependence of "b" and [V.sub.s]/<[V.sub.o]> on the shear stress in both cases would reflect the sensitivity of slip and shear flow to the polymer architecture in terms of "p" and "n" as explained below.

The sensitivity of "b" or [V.sub.s]/<[V.sub.o]> to the "p" and "n" values is the result of the influence of the molecular structure on these quantities. It is known that "n" decreases as the polydispersity and degree of long chain branching are increased and the molecular weight decreased, while the "p" value increases along with the polymer molecular weight [see, for example, Hatzikiriakos and Dealy (7)]. Therefore, many combinations may occur depending on the polymer characteristics. For instance, for a polymer with a high molecular weight and a narrow molecular weight distribution, "n" is close to one and "p" is large, so slip will be predominant and more likely to be observed. So, "b" and [V.sub.s]/<[V.sub.o]> will tend to increase along with the shear stress. However, if the polymer is polydisperse and/ or the molecular weight is low enough, "n" and "p" will be small, and then shear thinning will tend to mask or suppress the influence of slip. Therefore, "b" and [V.sub.s]/<[V.sub.o]> should decrease with increasing shear stress.

It is important to say that "p" could depend on each specific interaction between the polymer and die wall material; such interaction, however, is still far from being understood. Anyway, for a given polymer-die pair, [E.sub.q] 7 should hold once slip is triggered.

It is interesting to note, in addition, that any increase or decrease in the ratio [V.sub.s]/<[V.sub.o]> will be also reflected in a corresponding change in the ratio [V.sub.s]/<V> in the same direction, because:

[V.sub.s]/<V> = [V.sub.s]/[V.sub.o] + [V.sub.s] = [V.sub.s]/[V.sub.o] (1/1 + [V.sub.s]/[V.sub.o]) (8)

Thus, the ratio [V.sub.s]/<V> can also be used to measure the influence of slip on the whole flow field. Observe that both [V.sub.s]<[V.sub.o]> and [V.sub.s]<V> make more physical sense than "b" and can be used to measure the increase in productivity during the extrusion with slip.

In order to illustrate the given reasoning and analysis, in the following we present experimental results obtained with two LLDPEs, in which the previously two possible described behaviors can be observed.

MATERIALS AND METHODS

The polymers used were two linear low-density polyethylenes with a relative density of 0.918 and a MFI of 1.0, one (that we will call A) produced by Aldrich [Cat. 42,807-8] with 5%-l0% hexene as comonomer and the other (called B) by Union Carbide [GRSN-7047] with butene as comonomer. Full molecular characterization of polymer B is found elsewhere (15). The polymer A was provided in pellets and is reported with no-slip or antiblock additives while B was provided in granules without additives.

Flow data were obtained in experiments carried out in single-screw extruders at T 200[degrees]C (15, 16). The pressure at the capillary entrance was measured with a pressure transducer whose variation under steady flow conditions was smaller than 0.3 MPa. The utilized capillaries were those previously employed by Ghanta et al. (14), one made of 464 naval brass and another of 304 stainless steel, both having a L/D = 20 and D = 1 mm. Such a L/D ratio is enough to make end effects negligible.

Flow enhancement in the brass die was promoted by cleaning it with purges (Epipurge[R], from Pellets International and Unipurge[R], from Union Carbide, respectively) before starting the experiments, which were run under a continuous flow of nitrogen supplied at the hopper in the feeding region of the extruder. The use of the purges in the stainless steel die was found to be irrelevant for the flow behavior.

RESULTS AND DISCUSSION

The flow curves for polymers A and B obtained with the stainless steel and brass capillaries are shown in Fig. 1 and 2, respectively.

It can be easily seen that the flow curves depend on the capillary materials and that those obtained with the brass die exhibit a flow enhancement or slip when they are compared to the stainless steel ones. Using stainless steel, tungsten carbide, aluminum, and glass capillaries we have previously observed negligible slip with LLDPE (17, 18) at stresses below the critical for the onset of the stick-slip behavior.

The level of sharkskin observed in the extrudates obtained with both dies was very similar at low shear stresses, but the defect disappeared in the brass die and became more severe in the stainless steel die, respectively, as the shear stress was increased. On the other hand, elimination of the unstable stick-slip region in polymer B and not in A is due to the presence of additives in the polymer and their interaction with the brass die (15).

The slip velocity in the brass die can be calculated by comparison of the flow curves obtained when using the stainless steel and brass dies (Figs. 1 and 2), from the differences in the volumetric flow rate for a given shear stress. Then, the slip velocity for a certain shear stress in the brass die is given by:

[V.sub.s] = 4([Q.sub.B] - [Q.sub.s-s])/[pi][D.sup.2] (9)

where [Q.sub.B] and [Q.sub.s-s] represent the volumetric flow rates for the brass and stainless steel dies respectively (here we have changed the sub-indexes "exp" and "O" for "B" and "s-S", denoting brass and stainless steel, respectively). The calculated slip velocity in the brass die using only data in the stable flow regime for both polymers is plotted in Figs. 3 and 4 as a function of the shear stress.

The straight lines observed in the log-log plots in Figs. 3 and 4 show that the slip velocity follows a power-law behavior as a function of the shear stress in both cases (as previously assumed). A deviation from the straight line occurs for polymer B because of the influence of the extensional flow at the onset of melt fracture. This deviation is due to an additional increase in the flow rate usually observed at the onset of the gross melt fracture when using flat entry dies (17). Benbow and Lamb (11) reported that no extensional flow but slip was the relevant variable for melt fracture in LDPE. Stronger adhesion was suggested in a brass die than in the steel one in that case. Although we have observed an increase in the shear rate for the onset of gross melt fracture in LDPE and polypropylene through brass dies (unpublished work), we have not performed a detailed study of the influence of die materials on the onset of gross melt fracture. Nevertheless, from Figs. 1 and 2 it is evident that the die material is relevant for the onset of the stick-slip instability, which provides evidence for it as an interfacial effect.

The corresponding slip power-law behavior for polymers A and B are given by Eqs 11 and 12, respectively:

[V.sub.sA] = 38.55[[tau].sup.1.75.sub.w] (cm/s) (10)

[V.sub.sB] = 619.44[[tau].sup.3.59.sub.w] (cm/s) (11)

On the other hand, the shear flow behavior (free of slip) for both polymers in the stainless steel die may also be well approached by a power law in the shear rate range prior to the onset of the stick-slip instability in Figs. 1 and 2. The corresponding power laws for polymer A and B, respectively, is given by:

[T.sub.wA] = 0.0275 [[gamma].sup.0.42] Mpa (12)

[T.sub.wB] = 0.0197 [[gamma].sup.0.47] Mpa (13)

With the above equations for the slip and shear flows, the "b", [V.sub.s]/<[V.sub.o]>, and [V.sub.s]/<V> values can be obtained for both polymers at a given shear stress. The corresponding plots as functions of the shear stress are shown in Figs. 5 and 6 for polymers A and B, respectively.

It is interesting to note that for polymer A, "p" = 1.75 and "n" from the stainless steel capillary data is 0.42, which gives 1/n = 2.38. Thus, "p - 1/n" = -0.63, and therefore "b" and [V.sub.s]/<[V.sub.o]> should decrease as the shear stress is increased, in agreement with the results displayed in Fig. 5. In the same way, for polymer B, "p" = 3.59 and "n" = 0.467, which gives 1/n = 2.14 and "p - 1/n" = 1.44. Hence, "b" and [V.sub.s]/<[V.sub.o]> should increase as the shear stress is increased, again in agreement with Fig. 6. It can be seen in addition that [V.sub.s]/<V> also follows a power law behavior for the shear stress range studied. Such behavior however, is expected to change, in agreement with Eq 8, as the shear stress is further increased.

Some results by other authors may be explained in the same way. For example, for the high-density polyethylene A in Hatzikiriakos and Dealy (7), n = 0.44. which gives 1/n = 2.272, and at T = 200[degrees]C, p = 2.95. So, "p - 1/n" = 0.677 and therefore "b" increases along with the shear stress as calculated by Wang [Fig. 8 in (6)].

Hatzikiriakos and Dealy (7) reported that polydispersity in HDPE (consistent with a small "n" value) suppressed slip. A similar effect would be produced by branching; this is perhaps one reason why branched polymers like low-density polyethylene do not exhibit measurable slip through conventional dies. Yang et al. (8) reported slip and a decrease in "b" for low-density polyethylene in fluoropolymer coated dies, which was attributed to shear thinning. This result is also in agreement with the analysis presented in this work.

Before concluding, it is convenient to mention some possible sources of error that can arise when evaluating the slip velocity using capillary rheometers. Awati et al. (19) have shown that viscous heating and pressure effects on the melt viscosity may lead to errors in the slip velocity determination. However, the evaluation of slip effects at shear rates comparable to those found in most polymer processing operations are difficult to perform in rheometers other than pressure-driven ones. In addition, the flow generated in a capillary rheometer is very similar to that found during many transformation processes of polymers, the results obtained using this sort of rheometer being of potential practical importance.

CONCLUSIONS

The slip phenomenon in the capillary flow of linear polyethylenes was analyzed in this work. An equation derived from a macroscopic analysis was obtained, which allows a description of the trend of the slip length and the ratio between the slip velocity and the free of slip average fluid velocity as the shear stress is increased. Two possible cases of increasing and decreasing "b", [V.sub.s]/<[V.sub.o]> and [V.sub.s]/<V>, were illustrated from data obtained In the continuous extrusion of LLDPE under strong slip. The increase or decrease in "b", [V.sub.s]/<[V.sub.o]> and [V.sub.s]/<V> is determined by the polymer structure through its shear-thinning behavior and "p" value.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

ACKNOWLEDGMENTS

This work was supported by CGPI-IPN (010565) and CONACYT (34971-U). The authors are COFAA-EDI fellows.

REFERENCES

(1.) M. Reiner, J. Rheol., 2. 338 (1931).

(2.) M. Mooney. J. Rheol., 2, 210 (1931).

(3.) M. M. Denn, Annu. Rev. Fluid Mech., 33, 265 (2001).

(4.) N. El Kissi and J. M. Piau, "Slip and friction of polymer melt flows," in Rheology for Polymer Melt Processing, 357, edited by J. M. Piau and J. F. Agasant, Elsevier Science B. V (1996).

(5.) F. Brochard and P. G. de Gennes, Langmuir, 8, 3033 (1992).

(6.) S. Q. Wang, Adv. Polym. Sci. 138, 239 (1999).

(7.) S. G. Hatzikiriakos and J. M. Dealy, J. Rheol., 36, 703 (1992).

(8.) X. Yang, H. Ishida, and S. Q. Wang, J. Rheol., 42, 63 (1998).

(9.) V. Mhetar and L. A. Archer, Macromolecules, 31, 6639 (1998).

(10.) Y. M. Joshi, A. K. Lele, and R. A. Mashelkar, J. Non-Newtonian Fluid Mech., 89, 303(2000).

(11.) J. J. Benbow and P. Lamb, SPE Trans., 3, 7(1963).

(12.) A. P. Metzger and C. W. Hamilton SPE Trans., 4, 107 (1964).

(13.) A. V. Ramamurthy, J. Rheol., 30, 337 (1986).

(14.) V. G. Ghanta, B. L. Rise, and M. M. Denn, J. Rheol., 43, 435 (1999).

(15.) J. Perez-Gonzalez and M. M. Denn, Ind. Eng. Chem. Res., 40, 4309 (2001).

(16.) J. Perez-Gonzalez, J. Rheol., 45, 845 (2001).

(17.) J. Perez-Gonzalez, L. Perez-Trejo, L. de Vargas, and O. Manero, Rheol., Acta., 36, 677 (1997).

(18.) L. Perez-Trejo, Acerca de la determination de los estados estacionarios en el flujo de polietilenos fundidos a traves de capilares, Master's thesis, Instituto Politecnico Nacional, Mexico (1999).

(19.) K. M. Awati, Y. Park, E. Weisser, and M. E. Mackay, J. Non-Newtonian Fluid Mech., 89, 117 (2000).

JOSE PEREZ-GONZALEZ (*)

(*) Corresponding author. E-mail: jpg@esfm.jpn.mx
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Author:Perez-Gonzales, Jose; Vargas, Lourdes De
Publication:Polymer Engineering and Science
Article Type:Abstract
Date:Jun 1, 2002
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