Measurement of apparent extensional viscosities of polyolefin melts from process contraction flows

J. Non-Newtonian Fluid Mech. 92 (2000) 203–226

Measurement of apparent extensional viscosities of polyolefin melts from process contraction flows M.T. Martyn, C. Nakason1 , P.D. Coates∗ IRC Polymer Science & Technology, Department of Mechanical & Medical Engineering, Bradford University, Bradford BD7 IDP, UK Received 3 December 1999

Abstract The extensional characteristics of branched and linear polyolefin melts have been evaluated in slit die flow cells with abrupt contraction ratios of 4:1 and 15:1 on a single screw extruder. Apparent extensional viscosities of the melts in planar flows have been obtained by two routes, the first employing extensional strain rate data measured from particle velocimetry, the second using a continuum mechanics analysis based on the entry flow profile. The influence of flow geometry on apparent extensional viscosity of the polymer melts has been investigated. The measured in-process apparent extensional viscosities of the branched and linear polyolefin melts are found to be in good agreement, despite differences in the strain histories imposed by two contraction geometries (centre line extensional stress and corresponding average axial strain rates differ). Particle velocimetry was found to be better than the continuum mechanics approach in obtaining extensional strain rates due to a lower susceptibility to experimental errors. It appears that the in-process methods of assessing apparent extensional viscosity, with an abrupt 180◦ entry slit die, are geometry independent for the range of materials, strain and strain rates covered and provide a useful technique for ranking process-typical extensional behaviour of melts. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Extensional viscosity; Extensional strain rate; Continuum mechanics; Flow visualisation

1. Introduction The extrusion of molten polymer through dies is an established industrial processing operation that has evolved over the past century. Early studies of the extrusion process have highlighted nonlinearity between flow rate and die pressure drop, extrudate swell on exiting the die, large entry pressure losses and the occurrence of instabilities of certain melts above critical extrusion rates. It has become increasingly clear that accurate modelling of these processing characteristics cannot be achieved using shear data alone ∗ 1

Corresponding author. Fax: +44-1274-234-505. Present address: Prince of Songkla University, Thailand.

0377-0257/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 0 2 5 7 ( 9 9 ) 0 0 1 0 9 - 3

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but requires an understanding of melt elasticity and extensional response. The majority of polymer conversion processes, in particular blow moulding, vacuum forming and film extrusion involve considerable extensional or stretching flows. Extensional flows will clearly be encountered in any conversion process in which forced convergent and divergent flows occur, such as in the entry region of extrusion dies and in injection nozzles. Knowledge of the process-typical extensional flow behaviour of polymer melts is a fundamental requirement for optimum processing and process simulation. 1.1. Laboratory extensional rheometry The non-Newtonian response of polymer melts in shear fields is well understood. However, the extensional response of polymer melts can be quite complex. Measurement of extensional properties of melts is a non-trivial task. Precise measurement of extensional viscosity can only be accomplished under conditions of constant rate of strain — a condition that is difficult to attain in practice for a polymer melt. Test methods capable of imposing steady extensional strain rates on polymer melts are of interest to polymer melt rheologists. Two instruments imposing different modes of free-surface deformation (melts without support), under well defined conditions have evolved. One instrument, developed by Munstedt [1], imparts a constant tensile stress on the melt during elongation using cams. The second instrument, in its original design, subjected cylindrically shaped samples to a constant stretching rate by means of rotating clamps. The latter apparatus, which is attributed to Meissner [2], has evolved as a commercial instrument, the Rheometrics RME extensional rheometer. The drive belt and clamp mechanism of the instrument is capable of imposing an exponential increase in stretch velocity to provide a theoretically constant strain rate on samples of rectangular section. Such an instrument enables the extensional characteristics to be evaluated at relatively low extensional strain rates but at Hencky strains up to ∼7. Early studies using the above instruments have demonstrated the low strain rate extensional response of melts is highly dependent on molecular topology. In particular, studies by Wagner [3], Laun [4] and Munstedt [5], Laun and Schuch [6], Ide and White [7], Koopmans [8] have shown that linear and branched melts respond quite differently when subjected to extensional stresses. The extensional behaviour of linear polyolefins is often characterised by an extensional viscosity thinning response when subjected to gradually increasing extensional strain rate, a behaviour which parallels the shear response common to most polymer melts. By contrast, branched polyolefins exhibit initial viscosity hardening followed by a thinning response when subjected to extensional flow fields [9,10]. Recent studies, however, suggest that the strain hardening characteristic, often quantified by a strain-hardening-parameter (SHP, the rise in extensional viscosity with time at constant stretch rate), is not only peculiar to polymers with branched architecture but has been observed for HDPE [11] and polystyrene [12]. It is generally accepted that a steady state, or plateau, extensional viscosity exists at low extensional rates, typically <10−3 s−1 , for HDPE and LDPE melts. This plateau tensile viscosity, ηext , is related to the zero shear rate viscosity, η0 , by the Trouton ratio, ηext =3η0 . At higher rates, between 0.01 and 1 s−1 , extensional viscosities are noted to increase to a maximum, which is more modest for HDPE’s, before decreasing at progressively higher rates. 1.2. In-process extensional rheometry The experimental techniques mentioned above involve free-surface deformation of melts — a condition which is applicable to a minority of polymer conversion operations. The constant strain rate criteria

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provides an idealised condition for measuring steady state and time dependent extensional viscosities. The Hencky strains, strain rates and the free surface conditions used in such off-line instrumentation are often not representative of real process conditions. The majority of processes involve the forced flow of melts through dies with complex forms where the melt stream is fixed by the profile of the flow channel surfaces giving rise to complex extensional strain fields. Although apparently simple, the flow of viscoelastic melts through abrupt axisymmetric contraction geometries encompasses most elements required for developing an understanding of the processing behaviour of polymer melts. In particular, measurement of die entry pressures provides a method that has been exploited in the assessment of extensional properties. Metzner [13] and later Cogswell [14] estimated extensional properties of polymer melts by applying continuum mechanics analyses to separate entry flow into shear and extensional components in the convergent flow regions encountered in abrupt entry contractions. The analyses assume plug flow of plane sections in the convergent flow region. Tensile stress and extensional viscosity can be calculated from knowledge of the shear viscosity and pressure drop in a convergent section. Convergent flow models have, therefore, been exploited in the derivation of extensional viscosities. Huang [15] proposed equations to calculate extensional viscosity for converging flow at the die entry by deriving an explicit relation between the pressure drop and the normal stress in a convergent flow, starting from the equations of motion but making simplifying assumptions. Kwag [16] assessed Cogswell’s analysis comparing predicted entry pressure, stretch rate and extensional viscosities of linear and branched polyolefins with properties determined from finite element methods. It was concluded that although Cogswell’s method is inaccurate it is still an acceptable approximate method for determining extensional viscosities at stretch rates above 10 s−1 . Gibson [17] refined the plug flow assumption of Cogswell’s analyses proposing a spherical velocity field model to describe convergent flow. Apparent extensional viscosities were generally found to be higher with the spherical velocity field model [18]. Bersted [19] also evaluated and refined the assumptions made in the converging flow analysis of Cogswell. In particular, the model assumed a power law fluid in shearing flow and variable extensional viscosity in convergence to a zero length capillary. This rigorous analysis of the continuum mechanics convergent flow produced closer agreement to data obtained from free-surface experiments. Later, Chohan [20] employed Cogswell analysis together with the approach of Huang [15] to calculate extensional viscosity of branched polyethylenes, concluding that the methods of Cogswell and Huang produce similar results. Binding [21] proposed a theoretical analysis for convergent flow using energy principles to relate pressure drop to flow rate and fundamental rheological properties. The analysis relies on numerical solution and adopts a more rigorous treatment of flow through an abrupt geometry than the previous analytic models. More recently, both Muller et al. [22] and Padmanabhan [23] compared extensional viscosities derived from the Cogswell and Binding analyses and concluded the Cogswell analysis over-estimated the extensional viscosity. Moreover, the Cogswell model predicted lower strain rates and smaller natural entry profile angle than the Binding model. Assessment of extensional viscosities of melts under process histories was carried out in our laboratories by Groves et al. [18] using convergent flow pressure data. Comparisons were made between three continuum mechanics analyses of convergent flows, i.e., spherical flow field, planar flow field and minimum pressure models. Extensional viscosities of branched and linear melts were obtained by measuring entry pressure and shear viscosity data generated from off-line capillary rheometry and an in-process nozzle rheometer fitted to an injection moulding machine. Extensional data from the minimum energy model were shown to be consistent between both experimental techniques.

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1.3. Extensional rheometry comparisons In flows through convergent geometries, melts are subjected to progressively increasing stretch rates. There may also be uncertainties in strain history, for example upstream of the convergence. Previous research [3–7] has demonstrated that strain history has a significant effect on the magnitude of the measured extensional viscosities. Continuum mechanics analyses are, therefore, often queried because they are based on mean strain rates taken over the convergent section and, as such, derived extensional viscosities are likely to be an artefact of the process conditions and geometries used in the test method. There are, however, a number of studies that compare, and show consistency between, measured extensional viscosities of melts obtained from other methods involving extensional flow and convergent flow data. Schroff et al. [24] determined extensional viscosities from free surface isothermal melt spinning experiments. There was good quantitative agreement between extensional data from each method for LDPE, HDPE, PP and PS. There was also consistency between measured extensional viscosities of a polypropylene homopolymer melt in similar studies [25,26] using free surface melt spinning and forced convergent flow techniques. However, in a more recent, wider study comparing free surface stretching with convergent flow methods, considerable differences were reported in data from the same free surface method as well as data obtained from the two methodologies [27]. Results published on polyolefins evaluated from off-line, constant strain rate instruments indicate that their extensional behaviour is very sensitive to the strain history [3–7]. Typically, branched polyolefins exhibit a transient stress growth with time that is more prominent at high strain rates. Such characteristics pose additional complications when interpreting extensional flow derived off-line data to real melt processing. In real processing, time scales, strain rates and thermomechanical processing effects differ considerably from those of precision controlled laboratory test conditions of the constant strain rate devices. Knowledge of low strain rate, steady state, extensional viscosities of melts to very high strains is important for development of understanding of polymer rheology, but is not so relevant to actual processing of such melts through complex geometry dies. Important practical issues include: how stress grows at high extensional rates in typical process strain fields; and how extensional viscosity varies with process-typical strain rate and strain histories. Given the difficulties in interpreting and applying extensional data derived from steady state, free form, flow fields to real processing situations we explore a practical approach, electing to characterise melt behaviour in real process extensional flow fields. In our studies, we have evaluated stress, strain and strain rate fields, measured under process conditions, for melt flows in a flow visualisation cell incorporating an abrupt entry slit die on a commercial scale extruder. In particular, we define the extensional viscosity of melts derived by the localised ratio of centreline principal stress difference to local extensional strain rate. Clearly such values of extensional viscosity are obtained for non-steady state flow conditions, that is, melts exposed to conditions of varying strain rate over a known convergence (strain) path. For this reason, we refer to extensional viscosities measured using the flow visualisation system as an apparent extensional viscosity. Our previous papers [28,29] cover the development and quantification of the natural entry profiles and centre line principal stresses of several branched and linear polyolefin melts. In this article, we report on the methods used to derive apparent extensional viscosities of these polyolefins using stress data previously generated and strain rates measured from two methods. In the first method melt velocities, and so strain rates, along the centre line of flow into a slit die are derived by particle velocimetry. In the

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second method, we assume plug flow in the convergent region and apply a continuum mechanics analysis to the flow using the natural entry profiles as boundaries of the flow. 1.4. Stress optical law and extensional flow It is recognised that molecular orientation is much greater in extensional flow fields than shear flow fields. The high level of orientation has provoked much debate over the validity of the stress optical law in extensional flows. Matsumoto and Bogue [30] and Muller and Froelich [31] measured on-line birefringence in isothermal and non-isothermal experiments in extension of polystyrene melts. Their results indicate linear stress optical behaviour of the melts at low stress levels but pronounced departures from linearity at higher stress levels. They concluded that the stress optical law is not valid for the value of stresses greater than approximately 2×106 Pa. Koyama and Ishizuka [32] also studied birefringence in LDPE melts during simple extensional flow at constant strain rate and temperature. They found the relation between stress and birefringence gave a straight line with slope 45◦ indicating the stress optical law to be valid within the stress range evaluated (103 –106 Pa). Talbott and Goddard [33] reached similar conclusion for polymer solutions with stress levels greater than 103 Pa. Muller [31] proposed reasons for the deviation from a linear relationship between stress and birefringence. They suggest that molecular orientation (and hence the observed birefringence) has a finite limiting value when the maximum extension of the molecular chains is approached while the associated stress tends theoretically towards infinity. 1.5. Influence of flow geometry It is acknowledged that the geometry of a die can have a significant affect on measured shear and extensional properties. The effect of slit aspect ratio, d/H (see Fig. 1), of a die on polymer melts flow was studied by den Otter [34]. It was concluded that wall effects of the flow are minimised when the aspect ratio is equal to or greater than 10:1. Han and Kim [35] studied the influence of die contraction ratio

Fig. 1. Schematic of a flow cell dimensions for a 180◦ abrupt geometry with the variable contraction ratio.

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of axisymmetric die geometries on the elastic property of polymer melts and concluded the property of melts to be independent of geometry above a critical contraction ratio of 12:1. Later, Han and Drexler [36] exploited both these findings and designed a slit die with contraction ratio of 12.8:1 and the aspect ratio of 10.6:1. They concluded that stress and velocity distributions obtained from such dies were geometry independent. Our previous studies covered flow and stress development of polymer melts in a flow cell with a 180◦ entry slit die of 10:1 aspect ratio and contraction ratio of 15:1. According to the work of den Otter [34], Han [35] and Drexler [36], these dimensions should provide planar and elastic invariant flow conditions with minimal wall effects. In addition to assessing the in-process measurement of extensional viscosity, we have broadened our studies to assess the influence of contraction geometry, therefore strain and strain rate history, on measured in-process extensional properties. This has been achieved by evaluating melt flow through two flow cells with 10:1 aspect ratio slits but having 15:1 and 4:1 contraction ratios. The melts are subjected to lower strains in these geometries than achieved by off-line instrumentation but still reflect process typical conditions. Measurement of stress, strain rates and extensional viscosities were performed assuming a 2D, planar flow by observing a representative plane of flow coincident with the central axis of the two flow cells. The primary reason for using a slit with a contraction ratio of 4:1 is that such a geometry has been widely used in the computer modelling [37–39] of the polymer melt flows.

2. Experimental details 2.1. Materials and equipment In this study, we have used two branched polyolefins, LDPE EXP2184 and LDPE XL422F, and a linear polyolefin HDPE 5050EA. Detailed rheological and stress optical characteristics of these materials and the flow visualisation system, imaging technique and stress analysis are reported in our previous papers [22,29]. The research reported here includes assessment of the in-process apparent extensional viscosity of melts and the study of the influence of die contraction ratio on the extensional characteristics of polymer melt flow. A slit die with contraction ratio of 4:1 was designed. A schematic of the die is shown in Fig. 1. Derived extensional flow data was compared with that obtained from the 15:1 contraction ratio cell used previously. Both flow cells had slit die geometries with 180◦ entries and slit aspect ratios of 10:1. The slit height, dimension H of Fig. 1, was fixed at 3.75 mm for the 4:1 die and 1.0 mm for the 15:1 contraction die. Melts are subjected to Hencky strains of 1.39 and 2.71, respectively, on flow through these cells. 2.2. Assessment of extensional strain rate by particle velocimetry Particle velocimetry was used to obtain the centre line velocities and velocity gradients of each melt at various flow rates. In this technique a trace amount of particulate, (150 mm silicon carbide), were introduced into the melt. Images of particles moving along the centre line were captured and recorded using a high speed video camera and video recorder. A NAC 400 high speed video camera enabled up to 200 frames/s of images can be captured as opposed to the standard 25 frames/s using normal CCD video cameras. Accuracy in computed velocities was markedly improved by using the high speed video camera. Typically at high flow rates and in regions of most interest close to the slit entry it was feasible to measure

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velocities of the order of 800±1.56 mm/s. In practice, for the flow geometries and flow rates used, such an accuracy was seldom required. It was not always necessary to analyse on a frame by frame basis. Software developed in our laboratories [40] allowed the selection of multiple frame intervals from the video recordings. Typically, every eighth frame was found to be adequate for the branched materials. Entry profiles and associated velocity profiles of linear melts tended to be more abrupt close to the slit entry. Consequently, accurate measurement of velocities of linear melt required a reduced frame interval.The displacements of an individual particle (x) in a known period of time (t) was monitored and captured along the centre line of the die using an image digitising system. Velocity (dx/dt) and velocity gradient (dv/dx=˙ε ) were determined along the centre line. 2.3. Assessment of extensional strain rate from entry profiles A simple continuum mechanics model was also used to derive centre line strain rates. In the model, we assumed simple extensional flow (plug flow) inside the natural entry profile (see Fig. 2) i.e. plane sections remain plane, and a single axial velocity applies to all points across the flow at a given axial position, x. This treatment assumes no shear flow occurs inside the natural entry profile. The value of this approach is that it provides a rapid and relatively easy measurement from which an extensional strain rate can be estimated. The volumetric flow rate, Q (m3 /s) is obtained from measured extruder output, kg/h, using melt density which was measured for each melt, for the chosen conditions, from PVT tests using a Rosand RH7 twin bore capillary rheometer. The mean axial velocity v xx at any axial position x is given by vxx =

Q , Ax

(1)

where Ax is the cross-sectional area inside the entry profile at position x. However, A = d 2y,

(2)

where y is the co-ordinate of the natural entry profile at axial position x, hence 2y is the ‘height’ of the natural entry profile at position x (Fig. 2) and d is the depth of the flow channel (here 10 mm).

Fig. 2. Schematic diagram showing methods used for assessing extensional viscosity.

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Therefore, average axial velocity (vxx ) at position x can be determined from vxx =

Q . d 2y

(3)

By definition, the axial extensional stain rate ε˙ xx is given by the velocity gradient, i.e. dvxx . (4) dx In reality there will clearly be shear flow components inside the entry profile. However, in strong convergences extensional effects dominate so existing shear effects are neglected in this simple model. ε˙ xx =

2.4. Apparent extensional viscosity calculation Apparent extensional viscosities (ηext ) can be determined from extensional stress (σ E ) and strain rate (˙ε ). In our study, apparent extensional viscosities were evaluated from the ratios of extensional stress (principal stress differences) and extensional strain rate obtained at points along the centre line of the melt flow, for each flow cell. Extensional strain rate was obtained by two methods described in Sections 2.2 and 2.3. The derivation of extensional viscosity using these two methods of strain rate assessment are shown schematically in Fig. 3. Knowledge of stress optical coefficients for each melt enabled values of extensional stress (σ E ) to be calculated from birefringence patterns at the intersection of each fringe order along the centreline of flow. The fringes represent loci of constant normal stress difference. On the centre line, there is a pure extensional stress. Centre line extensional stress profiles were produced by curve fitting the discrete values of extensional stress. Similarly, extensional strain rate profiles were also produced using data from the particle velocimetry or continuum mechanics methods. Values of apparent extensional viscosity (ηext ) were then derived using the ratios of local stress (σ E ) and strain rate (˙ε ) developed for the centreline. That is for a given position (x) on the centre line the apparent extensional viscosity is σE ηext = . (5) ε˙

Fig. 3. Schematic of a simple model used to calculate velocity gradient.

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3. Results and discussion 3.1. Measured extensional strain rate Extensional strain rate data for LDPE’s; EXP2184 at 200◦ C and XL422F at 190◦ C calculated from the particle velocimetry and natural entry profile methods are presented in Figs. 4 and 5, respectively. The calculated extensional strain rate data for HDPE 5050EA at 200◦ C are shown in Fig. 6. The extensional strain rate data presented were derived using the 15:1 contraction ratio die and a slit wall shear rate of 365 s−1 . There is good agreement between the two datasets for the axial strain rates for LDPE EXP2184. However, in the case of LDPE XL422F there is only reasonable convergence of the datasets to a distance approximately 2 mm upstream from the slit entrance. At closer proximity to the slit entry axial strain rates computed from particle velocimetry are greater than those from the digitised entry profile method. Similar findings are also noted between computed axial strain rates for HDPE 5050EA. It is clear that the shape of natural entry profile of this melt is concave and relatively abrupt resulting in a high gradient (dy/dx) in the area of most interest, i.e. near the slit entry. Consequently, small errors in measurement of the difference in axial distance (dx) lead to large errors in the computed velocity gradient (dv/dx) for this melt. The extensional strain rate field of HDPE was successfully calculated using particle velocimetry. By contrast, the natural entry profiles of the LDPE’s are much more gradual in the critical area, near a

Fig. 4. Natural entry profile and average axial strain rate calculated by two methods for LDPE EXP2184 at 200◦ C and slit wall shear rate of 365 s−1 .

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Fig. 5. Natural entry profile and average axial strain rate calculated by two methods for LDPE XL422F at 190◦ C and slit wall shear rate of 365 s−1 .

Fig. 6. Natural entry profile and average axial strain rate calculated by two methods for HDPE 5050EA at 200◦ C and slit wall shear rate of 365 s−1 .

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slit entry. Consequently, the accuracy in computed velocity gradients and extensional strain rates from this method is greater for these melts than the linear polymers. The observed differences in measured extensional strain rates derived from the digitised entry profile and particle velocimetry is to be expected. There are significant errors associated with our approximation of plug flow in the derivation of strain rates from the entry profile technique. In particular, the non-Newtonian behaviour of polymer melts promotes parabolic velocity profiles with the shape of the profiles strongly dependent on the shear thinning characteristics quantified by the power law index (n). Such a profile would exist in the upstream reservoir some distance from the slit and tend to be exaggerated as the melt progresses through the convergent section bounded by the natural entry boundary. Shear and extensional flow coexist in the entry region. A single axial velocity is not strictly applicable for the flow in this region since the shape of the flow front will become progressively parabolic. Moreover, as melt flows from the upstream reservoir into a slit the streamlines converge and accelerate, producing a strong and dominant extensional flow near the entry of the slit. However, in the area of prime concern, on the centre line, extensional flow will predominate. Ignoring the shear flow effects and assuming plug flow leads to erroneous strain rates. For these reasons, the particle velocimetry method provides a more accurate measure of the actual melt strain rates in the region close to the slit entry. Measurement errors may occur at the region of the natural entry profile remote from the slit towards positions C or D of Fig. 1. The errors in this region are associated with difference of transverse distance (dy) which are very small compared to the difference of axial distance (dx). Such errors are expected to have negligible affect on calculated extensional viscosities of the melts. It is concluded that of the two methods used particle velocimetry using a high speed video camera is most reliable and accurate method for determining the centre line extensional strain rates for these melts. 3.2. Influence of contraction ratio on isochromatic fringe patterns Isochromatic fringe patterns representing the loci of points of constant normal stress difference for flows of LDPE XL422F and HD5050 melts in the 15:1 and 4:1 contraction ratio slit dies at 190◦ C are presented in Figs. 7 and 8, respectively. It is noted that the isochromatic fringe patterns for these two melts are of similar form in the 15:1 contraction ratio slit. Contraction ratio is clearly found to affect the shape of the fringe patterns. The affect is rate dependent and particularly prominent for the LDPE’s. The isochromatic fringe patterns of the HDPE 5050EA melt in a 4:1 contraction ratio slit are quite similar to that of the 15:1 contraction geometry at flow rates less than 125 s−1 and take on a semi-elliptical form. However, at higher flow rates, typically >125 s−1 a bilobal fringe pattern develops symmetrically about the centre line of flow. Similar fringe pattern shape development occurs for the LDPE XL422F in the 4:1 contraction slit die. The fringe pattern progress from being semi-elliptical at low flow rates, typically equivalent to wall shear rates of 30 s−1 to a pronounced bilobal pattern at higher flows. Further differences were observed in the isochromatic fringe patterns of LDPE melts in the contraction corners of the two dies In particular, at high flow rates a number of fringes were noted to progress into the corner region (e.g. A or B of Fig. 1) for the 4:1 contraction die. The fringes appear to move into the recirculation zone. Such features were not observed for the 15:1 contraction die for any of the LDPE’s even at the maximum wall shear rates of 600 s−1 evaluated. There is, therefore, the possibility of obtaining stress distributions for the LDPE’s in the recirculation region of the 4:1 contraction ratio slit geometry.

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Fig. 7. Isochromatic fringe patterns for LDPE XL422F at 190◦ C at various slit wall shear rates in a slit with contraction ratios of 4:1 (a and b) and 15:1 (c and d).

Fig. 8. Isochromatic fringe patterns for HDPE 5050EA at 190◦ C at various slit wall shear rates in a slit with contraction ratios of 4:1 (a and b) and 15:1 (c and d).

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3.3. Influence of die contraction ratio on centre line principal stress difference Centre line principal stress difference (PSD) obtained for LDPE XL422F and HDPE 5050EA at 190◦ C for flow in 15:1 and 4:1 contraction dies at slit wall shear rate of 125 s−1 are shown in Figs. 9 and 10, respectively. Die contraction ratio clearly has little effect on the fringe order of the LDPE XL422F melt. Similar observations were made at a flow rate of 80 s−1 . The fringe orders are the same in both contractions at slit wall shear rates of 80 and 125 s−1 being 5 and 7, respectively. This is not the case for the HDPE 5050 melt. With this melt the isochromatic fringe order in a 4:1 contraction ratio slit was typically found to be one fringe order higher than in the 15:1 contraction ratio slit for the same wall shear rate. For the geometries evaluated, stress levels in a 4:1 contraction ratio slit are higher than in a 15:1 contraction slit with this particular grade of HDPE, for slit wall shear rates greater than 80 s−1 . It is also clear from the data of Figs. 9 and 10 that contraction ratio does affect the centre line PSD profiles. At a given axial displacement the values of PSD are higher for the 4:1 than the 15:1 contraction geometry for both melts. It is coincidental that values of PSD shown in Fig. 9 for LDPE XL422F at the slit entry for both contractions are the same for this flow condition. The isochromatic fringe patterns are loci of constant normal stress difference which represent both normal stress and the value of maximum shear stress at that position. The shift in normal stress difference profiles for the 4:1 contraction is attributed to differences in volumetric flows. At the same slit wall shear rate, the volumetric flow rate (Q) of the melts flow in a 4:1 contraction ratio slit is higher than the 15:1 contraction ratio slit. In this work, slit wall shear rates were calculated based on a derivation of Walters [41] as shown below:      6Q 2+β 2+β = γ˙app , (6) γ˙w = h2 w 3 3

Fig. 9. Centre line PSD for LDPE XL422F at 190◦ C and a slit wall shear rate of 125 s−1 , for two different die contraction ratios.

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Fig. 10. Centre line PSD for HDPE 5050EA at 190◦ C and a slit wall shear rate of 125 s−1 , for two different die contraction ratios.

where Q is the volumetric flow rate (m3 /s), h the height of a slit channel (m), w the width of a slit channel (m), γ˙app the apparent shear rate (s−1 ), τ w the shear stress at the wall of a slit and β=

d[log(6Q/wh2 )] . d[log(τw )]

(7)

Plotting log γ˙app versus log τ w , reveals the general behaviour of the function. In particular, power law behaviour is indicated if the relation results in a straight line with slope greater than unity and n=1/β. Therefore, Eq. (6) can be transformed to   2n + 1 γ˙w = γ˙app . (8) 3n The above equation has the same form as Rabinowitsch corrected shear rates, or the true shear rate (γ˙true ) for melt flow in a slit die. It is clear from the above derivation the volumetric flow rate (Q) in the 4:1 die contraction ratio slit is approximately 10 times higher than that of the 15:1 contraction die for the same slit wall shear rate. Therefore, it is to be expected that the centre line PSD profile would be higher in the 4:1 contraction ratio slit. To confirm this, we present centre line extensional stress data for melt flow at the same volumetric flow rate, of 4.6×10−7 m3 /s, for both die contraction ratios at 190◦ C for LDPE XL422F and HDPE 5050EA in Figs. 11 and 12, respectively. The data show that, at the same volumetric flow rates, centre line PSD for both LDPE XL422F and HDPE 5050EA in a 15:1 contraction ratio slit are considerably higher than that of a 4:1 contraction ratio slit. This is reasonable since the pressure drop across the slit

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Fig. 11. Centre line PSD for LDPE XL422F at 190◦ C and a volumetric flow rate of 4.7×10−7 m3 /s, for two different die contraction ratios.

Fig. 12. Centre line for HDPE 5050EA at 190◦ C and a volumetric flow rate of 4.6×10−7 m3 /s, for two different die contraction ratios.

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channel, and therefore shear and extensional stresses of the melts flow, in the 4:1 contraction geometry of 3.65 mm height is lower than that a 15:1 contraction ratio slit for this flow condition. It is recognised that the ranking of PSD results is specific to the geometries studied. The data reported here concerns our original flow visualisation cell. New cells have since been designed and are in use in current studies [42]; these have greater flexibility, e.g. providing the opportunity to alter flow width (up to 25 mm, for studies of 3D effects in flows) and reservoir area, which will influence the amount of elastic deformation incurred in that region. 3.4. Apparent extensional viscosity Apparent extensional viscosities of LDPE EXP2184 at 200◦ C, LDPE XL422F at 190◦ C and HDPE 5050EA at 200◦ C are shown in Figs. 13–15. The extensional strain rates have been calculated from particle velocimetry and entry profiles, for the 15:1 contraction, and flow conditions equivalent to a slit wall shear rate of 365 s−1 for each melt. It is clear from Figs. 13 and 14 that both methods for assessing the apparent extensional viscosities for these grades of LDPE produce reasonably consistent data. The correlation is expected based on the reasonably close agreement in extensional strain rates from these two methods as previously shown in Figs. 4 and 5. By contrast, in the data of Fig. 15 there is considerable discrepancy between values of apparent extensional viscosity derived from the two methods with the HDPE 5050, particularly at low strain rates <20 s−1 . We attribute this poor agreement to measurement errors in the digitised entry profiles used to derive the strain rates. There are less measurement errors using particle velocimetry so the derived are probably more representative of the low strain rate values. There is, however, closer agreement in

Fig. 13. Apparent extensional viscosity of LDPE EXP2184 at 200◦ C and a slit wall shear rate of 365 s−1 .

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Fig. 14. Apparent extensional viscosity of LDPE XL422F at 190◦ C and a slit wall shear rate of 365 s−1 .

Fig. 15. Apparent extensional viscosity of HDPE 5050EA at 200◦ C and a slit wall shear rate of 365 s−1 .

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the apparent extensional viscosity derived from each method at higher axial strain rates (50–100 s−1 ). The closer agreement is attributed to improved accuracy in the computed extensional strain rate from the entry profile analyses. The entry profile of the HDPE melt is much more clearly observable closer to the slit that in the upstream region, consequently measurement errors associated with digitisation method are reduced in the region close to the slit entry. 3.5. Influence of die contraction ratio on strain rate and extensional viscosity measurements Natural entry profiles and extensional strain rates for 15:1 and 4:1 contraction ratios at a common slit wall shear rate of 125 s−1 are presented in Figs. 16 and 17 for melts LDPE XL422F and HDPE 5050EA. The average extensional strain rates are calculated from digitised entry profiles for these melts at 190◦ C. In general, average axial strain rates are found to be higher for the 4:1 contraction than the 15:1 contraction geometry for the same wall shear rate conditions irrespective of the polymer, an exception being strain rates derived close to the slit entry for melt LDPE XL422F with the 15:1 contraction. As previously stated the volumetric flow rate in the contraction ratio of 4:1 die is more than 10 times higher than the 15:1 contraction to achieve the same slit wall shear rate. Therefore, at the same slit wall shear rate both average strain rates and centre line extensional stresses in the upstream reservoirs of a 4:1 contraction are expected to be higher than for a 15:1 contraction flow, as is reflected in Figs. 16 and 17. Apparent extensional viscosities (ηext ) of LDPE XL422F and HDPE 5050EA obtained from the digitised entry profile method for the same flow conditions are presented in Figs. 18 and 19, respectively. Given

Fig. 16. Average axial strain rate calculated by digitised entry profile method, for LDPE XL422F at 199◦ C and slit wall shear rate of 125 s−1 , for two different slit contraction ratios.

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Fig. 17. Average axial strain rate calculated by digitised entry profile method, versus axial position for HDPE 5050EA at 190◦ C and slit wall shear rate of 125 s−1 , for two different slit contraction ratios.

Fig. 18. Apparent extensional viscosity calculated by digitised entry profile method, versus axial extensional strain rate for LDPE XL422F at 190◦ C and slit wall shear rate of 125 s−1 , for two different slit contraction ratios.

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Fig. 19. Apparent extensional viscosity calculated by digitised entry profile method, versus axial extensional strain rate for HDPE 5050EA at 190◦ C and slit wall shear rate of 125 s−1 , for two different slit contraction ratios.

the differences in strain histories imposed by the two convergent geometries there is good agreement between these results. The measured, low strain (here up to Hencky strains of 2.71), apparent extensional viscosities appear to be geometry independent for each contraction for both linear and branched melts. This agreement encourages us to view the results as reflecting material parameters rather than geometry effects. This has been demonstrated for two distinct molecular feature polymers. Data presented in Fig. 20 compares the apparent extensional viscosities of LDPE EXP2184, LDPE XL422F and HDPE 5050EA at 200◦ C and slit wall shear rate of 365 s−1 derived from the particle velocimetry method applied to the 15:1 contraction die. For reasons already discussed, we elect to use extensional viscosity data based on strain rates derived from particle velocimetry. The viscosities of the melts are found to decrease with increasing strain rate over the range 0.1–100 s−1 and exhibit similar strain thinning characteristics to those previously reported in studies by Park et al. [37] and Barakos and Mitsoulis [43]. Moreover, other studies [15,19,20–22,44] applying Cogswell’s models [14] to LDPE and HDPE extensional viscosities show a similar decreasing trend for strain rates greater than 0.1 s−1 . It is clear from the above data measured apparent extensional viscosities of both grades of LDPE are higher than the HDPE 5050EA. Fig. 20 also includes extensional viscosities, as presented in our previous work [29], derived by applying Cogswell’s minimum energy model [14] to capillary rheometry data of each melt at 200◦ C. While there is poor correlation between the sets of data the ranking of apparent extensional viscosity is the same for both techniques, i.e. LDPE XL422F>LDPE EXP2184>HDPE 5050EA. The strains and strain rate histories of the melts are quite different for both techniques so differences in measured extensional viscosities is to be expected. In particular, Hencky strains are ∼5.4 for the axisymmetric flow conditions of the capillary rheometer.

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Fig. 20. Comparison of the apparent extensional viscosities of LDPE EXP2184, LDPE XL422F and HDPE 5050EA at 200◦ C. Open symbols are data from Cogswell’s model, closed symbols are from flow visualisation at a slit wall shear rate of 365 s−1 .

Fig. 21. Trouton ratio (calculated apparent extensional viscosity/measured shear viscosity) for LDPE XL422F, LDPE EXP2184 and HDPE 5050EA at 190◦ C.

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Extensional properties of polymer melts are known to be strongly dependent on molecular structure, molecular weight (M¯ w ) and molecular weight distribution (MWD). In particular, long-chain branching is found to have the most significant effect on the rheological properties of polyethylenes [20]. Comparing the molecular characteristics of the materials, see Table 1 in [29], LDPE XL422F has very similar M¯ w and MWD to that of HDPE 5050EA but exhibits a higher apparent extensional viscosity for the same flow conditions. From our study, extensional viscosity appears to be independent of molecular weight and distribution. Although chain branching was not quantified the results suggest the differences in extensional behaviour between the branched melts is associated with molecular structure with the probability that LDPE XL422F has a greater degree of long-chain branching than that of LDPE EXP2184. Trouton ratios calculated using extensional viscosities from the particle velocimetry method and the shear viscosity from off-line Rosand RH7 twin bore capillary rheometry are presented for the LDPE grades XL422F and EXP2184 and HDPE 5050EA in Fig. 21. In theoretical treatments of Trouton ratio shear viscosities corresponding to low strain rates within the linear viscoelastic region are assumed in the relationship ηext =3η0 . However, in this work the lowest strain rate was higher than 3 s−1 . Therefore, all data in Fig. 20 represent flow behaviour of non-Newtonian polymeric melts, and the Newtonian plateau viscosity η0 is not involved in the calculation of Trouton ratios. It is clear Trouton ratios are higher for the LDPE’s than the HDPE 5050EA.

4. Conclusions Particle velocimetry and digitised natural entry profile methods were successfully exploited to assess the in-process extensional strain rates and apparent extensional viscosities of LDPE EXP2184 and LDPE XL422F at Hencky strains up to 2.71. Measurement uncertainties in the natural entry profile lead to discrepancies between the strain rates and subsequent extensional viscosities obtained from both methods, in the case of the HDPE 5050EA melt at similar strains. Particle velocimetry offers the more accurate method of obtaining centre line extensional strain rates. Axial strain rates computed from digitised entry profiles of LDPE XL422F and HDPE 5050EA melt flows at 190◦ C were greater in the 4:1 than the 15:1 contraction geometry at a common slit wall shear rate of 125 s−1 . Similarly, the magnitude of upstream centre line PSD profiles for LDPE XL422F and HDPE 5050EA were greater in a slit with a 4:1 contraction ratio than in a 15:1 contraction for the equivalent slit wall shear rates. The converse was true when equivalent volumetric flow rates were used. Despite differences in measured centre line extensional stress and corresponding average axial strain rates, hence strain histories, imposed by the two contraction geometries, the in-process apparent extensional viscosities of these branched and linear polyolefin melts were found to be in good agreement. It appears that such in-process methods of assessing apparent extensional viscosity, with an abrupt 180◦ slit entry die, are geometry independent and provide a useful technique of ranking process-typical extensional behaviour of melts. The ranking of apparent extensional viscosities of LDPE EXP2184, LDPE XL422F and HDPE 5050EA obtained from the particle velocimetry method was found to be in good agreement with that obtained from applying Cogswell’s minimum entry method [14] to off-line capillary rheometry data. The magnitude of Trouton ratios of the LDPEs EXP2184 and XL422F were greater than that of HDPE 5050EA reflecting the dominant extensional properties of the branched molecular architecture, i.e., extensional viscosities are substantially greater than shear viscosities. Trouton ratios of all the melts,

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