The effect of molar mass distribution on extrudate swell of linear polymers

J. Non-Newtonian Fluid Mech. 152 (2008) 195–202

The effect of molar mass distribution on extrudate swell of linear polymers C.F.J. den Doelder ∗ , R.J. Koopmans Dow Benelux B.V., PO Box 48, 4530 AA Terneuzen, The Netherlands Received 4 December 2006; received in revised form 20 February 2007; accepted 4 April 2007

Abstract Extrudate swell is an important viscoelastic feature of polymer melt processing. Experimental work with linear polymers has identified extrudate swell to depend strongly on average molar mass and molar mass distribution. Unfortunately there was no method identified to distinguish the relative effects of average mass and polydispersity. This paper revisits previously published experimental data on high-density polyethylenes and proposes a model to understand the coupled effect of these two molecular variables. The model connects findings from double reptation models via the recoverable compliance to extrudate swell. The reported results provide support to the model and lead to a better understanding of the important role of high-mass fractions in polydisperse polymers. © 2007 Elsevier B.V. All rights reserved. Keywords: Extrudate swell; Molar mass distribution; Double reptation; Polyethylene; High-density polyethylene

1. Introduction Polymer extrusion is an important process in transforming polymer materials into products. Examples of products formed via extrusion are blown film, extruded sheet and blown bottles. All extrusion processes involve a transition from confined flow to free-surface flow. The change in boundary condition at extrusion die exits causes the extrudate to swell. Understanding extrudate swell is essential for controlling dimensions of the final product. For example, in extrusion blow-molded bottles their wall thickness needs to be sufficient to prevent mechanical collapse of the bottle, but small enough to prevent excessive material use. The extrudate swell phenomenon, typically defined for capillary dies as the ratio of extrudate diameter over die diameter, has been studied extensively over the past four decades. Early work by Schreiber et al. [1] and Metzner [2] was followed by significant progress in the 1970s, following contributions from Tanner [3] and others, e.g. [4,5]. There is a general consensus that extrudate swell is related to the viscoelastic nature of polymer melts. More specifically, the “memory” of the polymer leads to recovery of strain after removal of stress. Tanner [3] relates

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extrudate swell to the ratio of first normal stress difference and shear stress. At the molecular level, extrudate swell is associated to the recovery of polymer chains to a state where orientation and stretch are relaxed. Considering a steady state flow and focusing on model extrusion systems described by a cylindrical reservoir, a contraction, a straight capillary die, and a free surface, extrudate swell needs to be specified at a specific position from the die or as function of time. In such a set-up, extrudate swell depends on the extrusion system’s geometry (contraction ratio and angle, die length and diameter) and on the operating conditions (temperature, flow rate). Extrudate swell then becomes defined by the (mainly) elongational deformation in the contraction and die exit region, and by the shear deformation in the die [6]. Many studies have used a continuum mechanics approach to improve the understanding and prediction of extrudate swell [3,7]. The benefits to such an approach are the ability to avoid challenging and cumbersome experiments and to flexibly consider various extrusion systems via numerical simulation techniques, either as commercial software or as self-developed calculation routines. This approach became popular in the early 1980s, with early work by Crochet and co-workers [8] followed by many other studies, e.g. [9–13]. One very critical aspect in simulation is the accurate continuum mechanical description of the polymer melt. This strongly depends on the selection of

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an appropriate constitutive model [14]. Examples are the KBKZ [15] and Giesekus [16] models. Subsequently for a chosen constitutive model a methodology to determine the materialspecific parameters needs to be in place. The parameters are typically obtained from a combination of linear viscoelastic experiments providing a relaxation spectrum, and targeted nonlinear experiments, such as uniaxial extension, that define the non-linear parameters. The success of this approach depends critically on the ability of these rheological experiments to distinguish materials that are known to differ in extrudate swell. Koopmans [17] has shown that small “lot-to-lot” variation within a sample, which is hardly showing up in rheological experiments, can lead to a substantial difference in extrudate swell. In conclusion, it is difficult to achieve useful swell predictions using continuum models, in view of the difficulty to incorporate the details of molecular structure known to be of essential importance. In addition to the continuum mechanics approach, increased extrudate swell understanding has been obtained by experimental swell studies using materials of different chemical nature and molecular structure. The effect of average molar mass, typically expressed via the weight-average molar mass Mw , and of polydispersity has been studied among others by Graessley et al. [18], Rogers [19], Mendelson and Finger [20], Racin and Bogue [21], Rokudai [22], Dealy and co-workers [23,24], Koopmans [17,25], and Wang and co-workers [26,27]. This approach has generated various valuable data sets, each focusing on different aspects affecting extrudate swell. Generalization of the findings however is difficult as a consequence of the lack of independent control of the higher average moments of the distribution function of the polymers used, the uncertainty on their precise polymer architecture (e.g. linear versus branched polymers), and the sparse variation in the polymer characteristics per study. For example, early work by Rogers [19] mentioned an increase of extrudate swell with Mw , seemingly in contradiction with conclusions of Mendelson and Finger [20], who reported the opposite. Koopmans [25] showed that this was due to a different range of polymers (sparse variation), and united the two by showing the existence of a maximum extrudate swell at intermediate Mw . Still, in his approach the effect of Mw could not be fully separated from the effect of polydispersity. The objective of the present contribution is to consider available experimental data from a novel perspective, using insights from double reptation models as inspiration for a suitable polydispersity index to be used for examining how it defines extrudate swell. Simultaneously, the available experimental data on HDPE from Koopmans [25] are re-analyzed and expanded via non-traditional use of an emulator. These two new elements provide a new perspective on the relation between molecular structure and extrudate swell. In particular, they increase the understanding of the role of high-mass fractions in the molar mass distribution. Schreiber et al. [1] and Orbey and Dealy [28] suggested that adding a small amount of very long chains significantly increases extrudate swell. This was confirmed by Zhu and Wang [27]. The present approach quantifies this in terms of the recoverable compliance calculated from molecular structure via double reptation models.

2. Extrudate swell model Accurate extrudate swell models will take into account the conditions of the process under consideration. Effects of geometry, temperature, and flow rate have been mentioned in the introduction. In addition, effects of gravity (sag) and post-die cooling may be important, depending on the time and distance after exiting, and the weight of the sample hanging under the position of observation. For a general purpose model, focusing on the effect of molecular structure at practical conditions, capillary rheometer extrusion of a circular strand (fiber) is considered here. In a benchmark set-up, effects from contraction, die land and exit, and mild sagging and cooling will be important. An example of an extrudate swell indicator related to such a set-up is the so-called S300 value [25]. This is obtained using an Instron capillary rheometer with a die L/D (length/diameter) ratio of 5.25 (D = 5.31 mm), operated vertically at 190 ◦ C with an apparent wall shear rate of 300 s−1 . The S300 extrudate swell indicator is defined as the percentage of extrudate swell: S300=(Dfiber − D) × 100%, where the fiber diameter Dfiber is measured 8 cm beneath the die, when the strand just reaches that point after being cut at the die exit. The latter piece of information is relevant to estimate the sag effect. For such a set-up, the dependence of S300 on average molar mass, say Mw , depends on a number of effects. Firstly, sag is primarily dominated by the zero-shear viscosity, which increases monotonically with Mw for linear chains. Thus, S300 increases with increasing Mw , since sag thins the extrudate more when the viscosity is low. At a certain viscosity, sag is negligible and for further increase of Mw sag will have no effect. Secondly, the larger the Mw , the more the chains can be deformed during stressing, and thus the more they can reorient after cessation of stress, leading to increasing S300 with increasing Mw . This is especially true at fixed apparent shear rate. Operating at constant stress will reduce the importance of this effect. Thirdly, the higher the Mw , the more time it will take for the sample to reach a given extrudate swell level. This was illustrated for the monodisperse samples of Zhu and Wang [27]. Thus, at a fixed observation position, this effect leads to decreasing S300 with increasing Mw . Summarizing, at constant polydispersity, S300 is expected to show a maximum at an intermediate Mw . It is more difficult to identify the phenomena that govern the effect of polydispersity at constant Mw . In general, polydispersity increases the elasticity (e.g. defined via the recoverable compliance) of the material. As extrudate swell is an elastic recovery process, increased elasticity is expected to increase extrudate swell. It is not evident how this is different for ultimate extrudate swell versus time-dependent extrudate swell. This depends on the elastic behavior at various (relaxation/retardation) time scales. Upon applying a given overall rate, a resulting overall stress is present in a polymer. However, the stretch and orientation of chains of different length will be different at that stress. If there are only chains of the same length (monodisperse), they will all have the same stress. Typical values are 0.1 MPa for many polymers. Upon including some chains of much larger mass, the overall stress will not increase a lot, but the stretch and orientation of those high-mass chains will be

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significantly higher. Those chains will contribute significantly to the extrudate swell upon stress release, but they may take a lot longer to reach their ultimate extrudate swell. Zhu and Wang [27] showed that this effect is indeed strongest at long times, but that there is already a significant contribution at shorter times. The presence of a matrix of shorter chains apparently accelerates the extrudate swell of the longer chains as if they “lubricate” the process. For more complex systems, with a continuous variation of chain length, the combined resulting extrudate swell is not easy to predict a priori. Based on the Zhu and Wang results [27] it is not evident to relate extrudate swell directly to the capillary flow curve. A new hypothesis is formulated, inspired by the elastic character of extrudate swell. The hypothesis is that ultimate extrudate swell correlates with the recoverable compliance, and that the practical (short time) extrudate swell indicator S300 does so as well. The latter is based on the Zhu and Wang result [27] that long chains already act at short times. Recoverable compliance is an indicator of elasticity, as it governs the terminal behavior of the storage modulus. It is assumed that there is a direct relation between this linear viscoelastic quantity and the nonlinear viscoelastic character of extrudate swell within a class of polymers [18–20,24]. The recoverable compliance is determined from constrained recoil, whereas extrudate swell is related to free recovery. These two phenomena depend in different ways on molecular structure, but it is expected that creep recovery is more closely related to extrudate swell than any other property measurable in a standard rheometer. A fundamental theory directly connecting structure to extrudate swell is not available. However, there are various fundamental theories that connect structure to basic viscoelastic quantities, such as the recoverable compliance. The hypothesis above that correlates the recoverable compliance to extrudate swell thus enables a connection between detailed molecular structure and extrudate swell for linear polymers. Den Doelder [29] has recently explored various fundamental models of the double reptation type [30–32]. The double reptation framework by Tsenoglou [30] and Des Cloizeaux [31] operates at the molecular level and provides blend rules to generalize viscoelasticity of monodisperse linear polymer chains to polydisperse molar mass distributions of arbitrary type. The various implementations of the double reptation model vary in the description of the monodisperse relaxation function kernel [29] and as a consequence predict a different dependence of the recoverable compliance on molecular structure. Den Doelder [29] has studied this dependence for two clearly different kernels: single exponential and modified time-dependent diffusion [31,32]. Using a very large set of distributions, the importance of average molar moments up to Mz+1 for the recoverable compliance was shown for both kernels. This indicates that this is a generic aspect of the double reptation formalism. Genetic programming was used to obtain accurate relations between compliance and such distribution moments, valid for a large variety of molar mass distributions. These relations are complex but essential for detailed understanding. Practically, for the distributions employed in the experiments discussed in this paper, a power-law reduction provides an

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initially sufficiently accurate description of recoverable compliance Je0 as a function of moments of the molar mass distribution: (Mz /Mw )(Mz+1 /Mz )2 PJ = 0, 0 GN GN  2  Mz+1 Mz , PJ ≡ Mw Mz

Je0 =

(1)

where PJ is defined as the polydispersity index for elasticity, monotonically related to the recoverable compliance via the plateau modulus G0N . The hypothesis relating recoverable compliance to extrudate swell, combined with Eq. (1), provide a connection between molecular structure and extrudate swell. Thus a model is proposed where extrudate swell S300 is a function of average molar mass and polydispersity in the following decoupled way: S300(MMD) = f (Mw ) × g(PJ ),

(2)

where MMD represents the entire molar mass distribution, f is a function that has a maximum and g is a function that increases monotonically. 3. Experimental data re-analyzed Model testing will initially be done using the designed data set of Koopmans [25]. This set consists of 13 blends of Ziegler–Natta catalyzed linear polyethylenes. The extrudate swell indicator S300 was measured for all blends. Since the 13 combinations (with two replicates) were created using a Box–Behnken experimental design, a statistical model was built that acts as an emulator of any experimental blend that lies within the boundaries of the design space. In other words, there is access to experimental data for an infinite number of polymer blends, as long as they are within that space. The design was constructed around three variables: the average masses of both components, MwA and MwB , and the weight fraction of the high-mass component B, w. This design enables direct calculation of the total Mw : Mw = (1 − w)MwA + wMwB .

(3)

This is the quantity that was used to unite the results of Rogers [19] and Mendelson and Finger [20] by Koopmans [25]. It is also possible to calculate the other moments from the moments of the original blend components. However, whereas this is easy for the 13 original blends, it is not straightforward to obtain these values for the emulated blends that can be anywhere in the design space. The emulated blend moments can only be calculated when there is a unique relation between the Mwi of a component and its other moments. To solve this issue, it is assumed that the individual components can be described well via a log-normal distribution. This reduces the issue to finding a relation between Mwi and Mni of a component, because all higher moments are completely determined from the log-normal distribution character. It will be shown that it is possible to derive such a unique relation between the Mwi and Mni moments of the individual components for these materials.

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Table 1 Measured moments of the blend components Component

Mw (kg/mol)

Mw /Mn

log Mw (log g/mol)

1A 2A 3A 1B 2B 3B

44 64 119 164 250 467

16.9 9.7 8.0 7.1 7.1 4.8

4.643 4.807 5.077 5.215 5.398 5.669

The moments of the blends are calculated from   Mw 1−w w , = ((1 − w)MwA + wMwB )) + Mn MnA MnB Mz Mw = (1 − w)MzA MwA + wMzB MwB ,

(5) (6)

and Mz+1 Mz Mw = (1 − w)Mz+1,A MzA MwA +wMz+1,B MzB MwB . (7) The log-normal distribution has the following property: Mz+1,i Mzi Mwi = = Mzi Mwi Mni

(8)

Insertion of Eq. (8) into Eqs. (6) and (7) shows that the higher moments of the resulting blends can be calculated from the Mwi and Mni of the blend components: Mz Mw =

Fig. 1. Measured relation between Mwi and Mni of the individual blend components used in the experimental design study of Koopmans [25].

3 3 (1 − w)MwA wMwB + , MnA MnB

Mz+1 Mz Mw =

6 6 (1 − w)MwA wMwB + . 3 3 MnA MnB

(9) (10)

Table 1 shows Mwi and Mni of the blend components used in the experimental design. The Box–Behnken was based on the 10 log-transposed M values. The determination of the moments wi from GPC is described in [25]. The design space is

Fig. 2 shows the moment ranges that can be reached for the blend within this design space, calculated from Eqs. (3), (5), (9) and (10). They clearly depend strongly on Mw . This needs to be considered when analyzing the extrudate swell results independently in terms of Mw and polydispersity.

44 kg/mol ≤ MwA ≤ 119 kg/mol,

4. Results

164 kg/mol ≤ MwB ≤ 467 kg/mol,

0.3 ≤ w ≤ 0.6.

(4)

Fig. 1 shows the experimental relation between Mwi and Mni of the blend components. There is indeed a direct connection between the two moments for this set of samples. To generate a large data set of points in the design space, the following strategy is used. The split w is allowed to vary continuously between 0.3 and 0.6. The Mwi of the individual components are generated on a grid of five by five values, equidistant on a log-scale. The Mni of the individual components are taken from Fig. 1 by piecewise linear interpolation. The resulting values are given in Table 2. Table 2 Moments of the emulated distributions Component

log Mw (log g/mol)

Mw /Mn

A1 A2 A3 A4 A5 B1 B2 B3 B4 B5

4.64 4.74 4.84 4.94 5.04 5.22 5.32 5.43 5.51 5.61

17.0 12.0 9.3 9.0 8.4 7.1 7.1 7.0 6.2 5.2

In the original paper of Koopmans [25], the extrudate swell indicator S300 is given as a function of MwA , MwB , and w: S300 = −9658.87 − 1058.6565 log(MwA ) +4405.794 log(MwB ) + 1971.07w +126.1071 log(MwA )2 − 380.263 log(MwB )2 + − 131.038w2 − 30.682246 log(MwA ) log(MwB ) + − 53.4286w log(MwA ) − 294.383w log(MwB ), (11) with the Mwi expressed in g/mol. The analysis of the previous section enables calculating S300 as a function of the moments of the resulting blends. The attention is first put on the effect of Mw . Fig. 3 shows the results for the original 13 Box–Behnken blends used to build the model, together with three times 25 points at the grid defined by five MwA and five MwB values at w = 0.3, 0.45, and 0.6. At first sight, the results prove that the function S300 (Mw ) goes through a maximum, as stated in the introduction. However, there is a large spread of data due to polydispersity. To separate the two effects, the value of w is varied at each MwA − MwB grid point in order to create isopolydispersity data sets. Using PJ as candidate polydispersity indicator, the results are shown in Fig. 4. The increase of S300

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Fig. 2. Molar mass distribution moment ranges covered by the blends.

Fig. 3. S300 extrudate swell vs. Mw ; original 13 Box–Behnken blends and grid blends atw = 0.3, 0.45, and 0.6.

Fig. 4. S300 extrudate swell vs. Mw ; original 13 Box–Behnken blends and iso-PJ blends.

with Mw is evident at the various PJ levels. Clear evidence for the decrease at high Mw is however not found. In fact, this apparent decrease is found to depend strongly on the low polydispersity of the blends in that region (cf. also Fig. 2d). The two bottomright points of the original Box–Behnken blends have PJ values of 177 and 203, which is additional proof of the polydispersity contribution to the S300 decrease. The decrease with Mw could still occur at higher values, but there is no information in the range of these blends that proves it. It is noted that there is still a significant spread of data at constant Mw and PJ . This is a consequence of the assumptions regarding the shape of the component distributions. Fig. 4 shows that S300 increases with PJ at constant Mw . To show this more clearly, the complementary plot of S300 as a function of PJ for various iso-Mw data sets is given in Fig. 5. At each Mw , the extrudate swell-polydispersity relation is monotonic with variability that clearly exceeds the uncertainty due to the assumptions. The number of data points available at high Mw

Fig. 5. S300 extrudate swell vs. PJ ; original 13 Box–Behnken blends and iso-Mw blends.

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is limited and this plot shows again that it is difficult to prove the decrease of extrudate swell with increasing Mw in the highMw range, although the extrudate swell values at 275 kg/mol are very low. It is noted that the exact form of PJ cannot be identified from these specific data. Similar plots as Figs. 4 and 5 can be obtained when using somewhat different combinations of moments. It is however found that inclusion of the Mz and Mz+1 moment is critical. Experiments on materials with more accurately characterized distributions are required to obtain more detailed information on the exact form of PJ and thus to enable evaluation of the different kernels in the double reptation models. For practical purposes, the functions f and g from Eq. (2) are proposed to be of the following form: f (x) = ax2 + bx + c

(12)

and g(x) = dx + e.

(13)

The resulting combined function needs one parameter less: S300 = (aMw 2 + bMw + c)(dPJ + e) a c e = bd Mw 2 + M w + PJ + . b b d

(14)

The reason that b rather than a is put in the pre-factor is to allow the case a = 0, which corresponds to the absence of a maximum in f. Simplification of Eq. (14) leads to S300 = α(βMw 2 + Mw + γ)(PJ + δ).

(15)

The parameters α–δ are obtained from least-squares optimization using 184 data points: the original 13 Box–Behnken blends, the 75 grid blends at three w levels, 62 iso-Mw blends, and 34 iso-PJ blends, all within the original design space. The resulting quantitative model is S300 = 6.17 × 10−4 (−2.44 × 10−3 Mw 2 + Mw +8.37 × 10−3 )(PJ + 1.16 × 103 ),

(16)

Fig. 6. S300 extrudate swell model response surface vs. Mw and PJ with the measured points.

with Mw expressed in kg/mol, valid for the input range of Fig. 2d. The statistical indicator R2 between measured extrudate swell (Eq. (11)) and predicted extrudate swell (Eq. (16)) for the 184 data points is 0.74. Fig. 6 shows the model response surface from Eq. (16) in 3D together with the measured points from Eq. (11). This model does show the decrease of S300 with increasing Mw (β < 0), but not with strong statistical significance. The Mw at maximum extrudate swell is independent of PJ in this model. The value is 205 kg/mol. 5. Discussion The concept of connecting extrudate swell to structure via the compliance was already used by Rogers [19] and Mendelson and Finger [20]. The functional form they used for the highmoment polydispersity index was different and they found no clear correlation using the higher moments. When using the PJ polydispersity index with their data, still no clear correlation is found. An important reason is that in their data the variation in PJ is very limited. The variation in Mw is much stronger and dominates the extrudate swell response. It is noted that the design space of their materials is such that Mw /Mn is strongly correlated to Mw . This is why Mw /Mn seems to give a better correlation with extrudate swell than polydispersity indicators using higher moments. A second reason for the lack of correlation between the higher molar moments and extrudate swell for those data sets lies with the experimental difficulty in determining the higher moments. It is possible that the real PJ varies much more than reported. This is precisely the reason why blending systems of known architecture provides more information [24]. Even if the higher moments of the individual components are not exactly known due to the same experimental issues, the variation in higher molar moments of the blends will largely depend on the blend ratio and lower molar moments of the components. The assumption of log-normality of the blend components may not be very accurate, but it enables to show the main effects of the different moments. Using distributions with asymmetric tails such that Mz+1 /Mz < Mz /Mw < Mw /Mn leads to lower PJ but the response surface will qualitatively be the same. The model of Eq. (16) is not claimed to be universally applicable, but it is constructed to provide more insight in the role of the full molar mass distribution for extrudate swell of linear polymers. As such, the model is very useful in designing new polymers. The polystyrene blend data of Racin and Bogue [21], the polyethylene blend data of Swan et al. [24], and the polybutadiene blend data of Zhu and Wang [27] are in line with the data presented here. In the latter, a small fraction of high M material is blended with 15 times lower M monodisperse material. As Zhu and Wang work at constant stress, some of the arguments for an increase of extrudate swell with increasing Mw are not valid anymore. When deriving a practical extrudate swell index from their data, the extrudate swell value at a constant time, e.g. 10 s or 100 s can be taken. Then an extrudate swell decrease with Mw is found for samples with the same polydispersity and an increase with polydispersity is found for the unimodal polydisperse sample and for the blends. The effect of polydispersity is

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Table 3 Measured component moments and calculated blend moments with measured extrudate swell data at 200 ◦ C from Swan et al. [24] Material

Mw (kg/mol)

Mw /Mn

Mz /Mw

Mz+1 /Mz

PJ

S thick (%)

S diam (%)

SCLAIR 2910 SCLAIR 56B 80% 56B 60% 56B 40% 56B 20% 56B

55.6 179 154 130 105 80.2

3.1 9.4 8.2 7.0 5.7 4.4

2.9 5.9 6.4 6.9 7.3 6.9

2.7 3.6 3.8 4.1 4.7 5.9

21 78 94 118 160 239

– 85 110 130 162 130

– 70 83 100 115 98

significantly stronger than that of Mw in the range studied there. It is not possible to calculate PJ of the blends as the polydispersity of the high M material is not given. The polyethylene blends of Swan et al. [24] can be treated in a similar way as the Koopmans data [25], although the design space is much smaller: combinations of two original materials in different weight ratios. The Mz+1 -moment cannot be calculated directly, as the original Mz+1 is not provided. When assuming that Mz+1 /Mz Mz /Mw = (17) Mz /Mw Mw /Mn for the original materials (log-normality), the resulting blend moments can be calculated up to Mz+1 . They are summarized in Table 3 together with the measured extrudate swell data. The extrudate swell is represented with two indicators: thickness and diameter swell, since an annular die was used. The relation between thickness swell and diameter swell is monotonic for these samples. It turns out that both extrudate swell indicators correlate strongly with Mz /Mw and the role of Mz+1 seems to be less important. This should be treated with care, as Mw decreases with increasing Mz+1 and again the effects of average mass and polydispersity cannot be fully separated for this small data set. It would be interesting to obtain data on more extensive blend systems to validate the functional forms obtained with the present approach. 6. Conclusions Extrudate swell is a complicated phenomenon that can be studied in many ways. Focusing on the relation between extrudate swell and molecular structure, the present analysis aims to bring more understanding regarding the effect of chain length variation and distribution for linear polymers. Since a full theoretical treatment of the dependence of extrudate swell on structure is not available, the emphasis here is put on maximizing the empirical information from clever experiments, while using guidance from available theories that connect structure to basic viscoelastic quantities. Previous experimental studies focused on sample sets with limited variability. The first main contribution presented here is the enhanced analysis of available blend data to create a data set with much larger and more systematic variation in molar mass distributions. The second main contribution is the connection of the extrudate swell data to relevant polydispersity indicators obtained from fundamental structure-rheology models, via the recoverable compliance. The double reptation model is used to

show the relevance of moments up to Mz+1 for the elasticity of polymer melts. Combined with the incorporation of average mass effects via Mw , an empirical model is constructed using the re-analyzed data. Without claiming universality of the derived equation, it provides a framework where available data from other studies is shown to fit into. The model clearly identifies the critical aspects of the molar mass distribution of linear polymers in relation to extrudate swell. It also provides guidance for performing targeted additional experiments on well-characterized polymers to further enhance both qualitative understanding and quantitative accuracy. In that respect it would be interesting to confront the results of such tests with the variations in compliance for the different kernels used in the fundamental reptation models. Specific new insights from the present approach are: • commonly used polydispersity indicators such as Mw /Mn are irrelevant for predicting extrudate swell, • the Mz+1 moment is essential in quantifying the role of polydispersity and showing the dominant effect of the high mass tail, • the apparently strong decrease of extrudate swell with increasing Mw in the high Mw range is primarily due to a decrease in PJ (these quantities are strongly coupled for blends in that range). It is important to find a balance between increasing fundamental understanding on the one hand and following a practical approach that optimizes predictive capability with the amount of information available on the other hand. In its present form, the model is actually being used in practice to design new materials with specific extrudate swell requirements in a way that was not possible before. Acknowledgements The authors appreciate discussions with John Dealy of McGill University and St´ephane Costeux of The Dow Chemical Company. They thank Dow Benelux B.V. for the opportunity to publish this work. References [1] H.P. Schreiber, A. Rudin, E.B. Bagley, Separation of elastic and viscous effects in polymer melt extrusion, J. Appl. Polym. Sci. 9 (1965) 887–892. [2] A.B. Metzner, Historical comments on stress relaxation following steady flows through a duct or orifice, Trans. Soc. Rheol. 13 (1969) 467–470.

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