Rhe Pressure Dependece

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The pressure dependence of the shear and elongationalproperties of polymer melts1

D.M. Binding, M.A. Couch, K. Walters*

Department of Mathematics, University of Wales, Aberystwyth, UK

Received 3 April 1998; revised 14 May 1998

Abstract

A capillary rheometer has been modified, by the addition of a second chamber and valve arrangement below the main die, in

order to measure the pressure drops associated with the capillary and entry flows of a number of polymer melts as a function of

pressure. The five polymer melts investigated are high- and low-density polyethylene, polypropylene, polymethyl-

methacrylate and polystyrene, each of which is tested at three temperatures within the normal processing range, at apparent

shear rates between 50 and 2500 sÀ1 and at mean pressures ranging from atmospheric up to 70 MPa. The capillary pressure

drop data are used to obtain shear viscosity functions using conventional capillary rheometry expressions, whilst extensional

viscosities are estimated from orifice pressure drop data via the Cogswell±Binding analysis. Both the shear and extensional

viscosity curves for all of the polymers are seen to exhibit an exponential pressure dependence that can be characterised bypressure coefficients that are found to be independent of temperature. Trouton ratios for the polymers can be specified by an

expression with separable strain rate and pressure dependence terms, the latter of which is again exponential. The pressure

coefficients of the Trouton ratio terms then orders the pressure dependence: PS>PMMA>PP>HDPE>LDPE. Our major

conclusion is that the Trouton ratio for some of the polymer melts can be a strong function of the pressure, indicating that the

variation of extensional properties with pressure can be greater than that of the shear properties. # 1998 Elsevier Science B.V.

All rights reserved.

Keywords: Polymer melts; Shear viscosity; Extensional viscosity; Contraction flow; Pressure dependence

1. Introduction

It is known that the viscosity of a wide range of liquids is a strong (exponential) function of pressure.Little is known of the effect of pressure on other rheometrical properties, such as normal-stressdifferences, dynamic moduli and extensional viscosity. High pressures exist in many industrial

processes. For example, in the polymer industry, processes such as injection moulding can involve

J. Non-Newtonian Fluid Mech. 79 (1998) 137±155

ÐÐÐÐ

* Corresponding author. Fax: +44-1970-622777.1Dedicated to Professor Marcel J. Crochet on the occasion of his 60th birthday.

0377-0257/98/$ ± see front matter # 1998 Elsevier Science B.V. All rights reserved.PII: S 0 3 7 7 - 0 25 7 ( 9 8 ) 0 0 1 0 2 - 5



pressures up to 100 MPa; the pressures experienced by lubricants in gears can often exceed 1 GPa;whilst oil-well drilling muds have to operate at depths where the pressure can be in excess of 20 MPa.

There is, therefore, an evident need to study how the viscoelastic and elongational properties of

industrial fluids vary with pressure. This paper will begin to address this need by studying the effect of pressure on the shear and extensional viscosities of polymer melts, using a suitably modified capillary

rheometer.

2. Previous work

Experimental investigations into the effect of pressure on the shear viscosity of polymer melts were

initiated in the late 1950s by Maxwell and Jung [1]. They looked at the effect of hydrostatic pressure onthe apparent viscosity of high molecular weight polyethylene (branched) and polystyrene, using whatwas essentially an opposed-barrel capillary rheometer. A capillary with a length to diameter ratio

(aspect ratio, L/D) of 20 was chosen to ensure some degree of laminar flow, but they made nocorrection for entrance effects which were assumed to be negligible. The apparent viscosity of

polyethylene increased 14-fold as the mean pressure was increased from atmospheric to 168 MPa,whilst polystyrene showed a remarkable 135-fold increase over the range 0±126 MPa.

A more comprehensive study was undertaken later by Westover [2], who was also interested in the

effect of pressure on the rheology of polyethylene. He developed a capillary rheometer, based onMaxwell's design, that was capable of operating at pressures up to 170 MPa, over a wide range of temperatures. As was the case with Maxwell's instrument, pressure drops were estimated by calculating

a force balance on the rams, since these devices were conceived before the advent of advanced flush-mounted pressure transducers. With this rheometer, Westover was able to measure the apparentviscosity of high- and low-density polyethylene (and to a lesser extent polypropylene and polystyrene)

over a respectable range of shear rates and at several levels of mean pressure between 14 and 170 MPa.In addition, he measured the effect of pressure on entrance flow, but sadly did not publish the details of

these measurements. What he did mention was that, for a capillary die having an L/D 20, this effectcould be 25±50% of the total pressure loss, depending on die size, polymer, temperature, flow rate and

pressure. He also looked at induced crystallisation at moderate temperatures.In the late 1960s, Choi [3] developed a capillary instrument from an extrusion rheometer that

consisted of a barrel and capillary in series. The barrel itself was used as the main capillary tube for

viscosity determination and several different capillaries, having L/D's of 3.5±100 were used to vary themean pressure from ambient up to 175 MPa. Therefore, the aspect ratio of the barrel (of diameter0.95 cm) could easily be varied from 3±15 by altering the melt height. Choi found that the apparent

viscosity of a polyethylene, tested at 1908

C and a shear rate of 7.12 sÀ1

, increased nearly four-fold overthe full range of pressure. He also examined the flow pattern in the barrel using coloured layers of meltas tracers and found regular laminar flow. However, this method was limited to low shear rate < 10 sÀ1,

and hence Choi, in conjunction with Nakajima, [4], later went on to consider another method for highershear rates (10±500 sÀ1), which involved observation of the non-linearity of Bagley-type plots at high

pressure. This capillary technique, which involved making refined calculations from the non-linearityin plots of pressure drop versus die length, had first been developed by Duvdevani and Klein [5] andwas later adopted by Penwell et al. [6] and Casale et al. [7]. Duvdevani and Klein studied the pressure

effect in the capillary flow of polyethylene using an exponential viscosity±pressure relation. Theyderived expressions for the shear stress at the die entrance and exit that were then employed to

138 D.M. Binding et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 137±155



determine the pressure coefficient via regression analysis. Casale et al. [7] looked at a single sample of polymethyl methacrylate for several capillary ratios varying from 4±100 at a number of temperatures

between 1608C and 2508C. They used a model derived from the WLF equation to pressure-correct the

capillary data, which then agreed well with data obtained at atmospheric pressure with a WeissenbergRheogoniometer. The instrument used in the latter two studies was an Instron capillary rheometer

capable of operating from 0±280 MPa over a wide range of temperature. Each of these studies requiredsensitive pressure transducers to measure the pressure gradients along the capillaries to a high degree of accuracy and assumed that entrance effects were independent of pressure.

The experimental work was continued in the early 1970s by Cogswell and McGowan [8] whoinvestigated the effects of both temperature and pressure on the viscosities of polymeric liquids.

Cogswell [9] later studied the effect of pressure on the apparent viscosity of polymer melts such aspolypropylene and high-density polyethylene. His main conclusions were that an increase in thepressure of a polymer melt was equivalent to a decrease in temperature, and that, if the viscosity was

very sensitive to changes in temperature, then a similar sensitivity to pressure could be anticipated. Inboth of these studies, the experiments were performed with a pressurised `Couette-Hatschek' rotating

cylinder viscometer similar in essentials to that developed by Semjonov [10±12].Using a piston-driven high-pressure slit-die rheometer, with three pressure holes evenly spaced along

the die and one in the barrel, Laun [13] was able to investigate viscosity, entrance and exit pressure

losses and the pressure coefficient of a high molecular weight LDPE melt. These results were comparedwith measurements on circular dies, assuming both linear and parabolic pressure profiles. He found thatthe true shear viscosity increased exponentially as a function of exit pressure, but no change was

observed in the entrance pressure loss.In 1990, Driscoll and Bogue [14] pioneered another experimental technique, similar to that of Laun,

for raising the mean pressure in a capillary die. Their method involved the addition of a downstream

chamber to an Instron capillary rheometer, which was held at a high pressure by means of a needlevalve. With this apparatus, they were able to measure the melt viscosity of a sample of polystyrene at

pressures of up to 124 MPa. Kadijk and van den Brule [15] also used a slit die for their study of thepressure-dependent viscosity of polystyrene, polyacrylonitrile-butadiene-styrene (ABS) and poly-

propylene. They determined temperature and pressure coefficients for each of the melts and concludedthat the pressure coefficients were independent of pressure for PS ad ABS, but decreased with risingpressure for PP.

More recently, Moldenaers et al. [16] considered non-linearities in the Bagley plots for thermotropiccopolyesters using both slit and capillary instruments. Kadijk and van den Brule's instrument andmethod were applied to measure the pressure dependence of the viscosity on mean pressure. They also

used a Gottfert 2002 capillary rheometer equipped with slits as well as capillaries to obtain Bagleyplots. They conjectured that the observed non-linearity in these plots arises, not only from the pressuredependence of the viscosity, but also from molecular-reorientation effects.

The development of a new capillary instrument capable of operating in both shear and oscillatorymodes at elevated pressures was recently described by Mackley et al [17]. Their instrument, known as a

`multi-pass rheometer', was similar in principal to that of Westover in that the fluid was containedwithin the capillary by two servo hydraulically controlled pistons. Mackley and Spilleler [18] used thisinstrument to study the pressure-dependent rheology of linear low-density polyethylene for pressures

up to 23 MPa. They discovered that both the apparent viscosity and viscoelastic data showed a linearincrease of about 20% over the pressure range tested.

D.M. Binding et al. / J. Non-Newtonian Fluid Mech. 79 (1998) 137±155 139



The effect of pressure on viscoelastic properties had earlier been studied by Ellis [19], who used anultrasonic technique to measure the viscoelastic relaxation of bitumens, mixtures and polymer solutions

in the high frequency range 5±78 MHz at pressures from 40±200 MPa. She found that the shear

modulus of all the liquids tested increased linearly with pressure at ambient temperature.

3. Experimental set-up

The instrument used to carry out the present experiments was a capillary rheometer originallydeveloped at the University of Wales Aberystwyth under a Department of Trade and Industry (DTI)

Link Scheme. This rheometer is capable of operating at pressures of up to 140 MPa and covers atemperature range from 20±5008C. It was, therefore, suitable for the present investigation.

Amongst the main features of the instrument, we mention: the instrument frame, a drive system

(consisting of a compact drive motor and gearbox) that controls the motion of the piston, a silent chaintransmission system with ball screws, and a stainless-steel twin-barrel assembly. Die ports are located

at the base of each barrel. In addition, a number of steel dies of diameter 0.5, 1 and 2 mm are suppliedwith the instrument. Temperature control is maintained by a heating unit, made up of a power supplyand three heater coils, wound separately around the barrel assembly at the top, centre and bottom. Each

coil is independently monitored by a precision platinum-resistance temperature probe (PT100), insertedbetween the barrels at the appropriate height, and thermostatically regulated by an electronic controlmodule. Reservoir pressures are measured by `Dynisco' pressure transducers (PT467E), flush mounted

on the wall of each barrel immediately above the extrusion die. The transducers are rated at up to140 MPa and are interfaced with the control system, which displays the reservoir pressures on theconsole. The control unit itself is a new T.A. Instruments Weissenberg Rheogoniometer controller

which has been adapted to conform to the rheometer electronics.A schematic diagram of the modified capillary rheometer is shown in Fig. 1. The main modification

is the addition of a second chamber, fabricated from forged steel and fitted below the die of one of therheometer barrels in place of the die-retention bolt. At the base of the chamber is a constriction made up

Fig. 1. Schematic diagram of the modified `high pressure' capillary instrument.




of a conical die and conical plug. The plug can be moved vertically by means of a screw thread in orderto vary the level of constriction in the die, thereby controlling the back pressure in the chamber. This

method of controlling the mean pressure is similar to that employed by Laun [13] and Driscoll and

Bogue [14], who also used valve arrangements to raise the mean pressure.An outlet hole, drilled immediately below the constriction, allows test liquid to exit the instrument.

The chamber has an internal diameter of 12.5 mm and length 130 mm, whilst the melt barrel is of diameter 25 mm and length 200 mm. A flush-mounted pressure transducer fitted at the mid-point of thechamber wall records the chamber pressure. This is also interfaced to the control unit. Temperature

control of the chamber is maintained by a separate system to that of the barrel. This consists of a heatercuff regulated by a thermocouple and thermostatically controlled power supply.

A crucial potential problem with the instrument concerned the possibility of leakage around the diesand past the piston, which would clearly distort the measurements of flow rate and pressure drop. Thepiston, therefore, was, modified to cope with the higher pressures generated by fitting a PTFE washer to

the brass head of the piston. As the pressure is applied to the end of the piston, the washer is squeezedout to form a tight seal against the barrel wall, since the end disc of the piston is free to slide onto the

washer. PTFE washers were also applied above and below the dies to ensure a good seal was madewhen the components of the modified instrument were screwed together.

Only one mode of operation is possible with the present instrument. A constant ram/piston speed can

be prescribed, which generates a constant flow rate throughout the instrument, once steady stateconditions have been realised. The constriction is then adjusted to control the pressure in the secondchamber. The upstream and downstream pressure transducers are used to measure the total pressure

drop associated with the die. At the same time, it is important to stress that the mean pressure within thedie itself cannot be controlled directly.

4. Experimental procedure

In the present study, we shall report on a number of commercial polymer melts. Five mainstream

polymer processing grades were chosen: High-density polyethylene, HDPE (BASF, Lupolen 1840H5431P); low-density polyethylene, LDPE (BASF, Lupolen 1840H); polypropylene, PP(ICI, GWM213)); polystyrene, PS (BASF, Polystyrol 143E) and polymethyl methacrylate, PMMA (ICI, CLH 374).

With the first four melts, no special preparation was required, although the PMMA required drying in avacuum oven prior to testing. Each of the polymers was tested at three convenient temperatures.

In the current study, two dies were employed: a capillary of diameter 1 mm and length 25 mm and an

orifice of the same diameter, but of nominally zero length. (The actual length was approximately0.25 mm). The mean pressures for all the polymers were limited to a maximum of around 70 MPa,although, at this mean pressure, the pressures in the barrel could be as high as 100 MPa. This was done

to avoid problems with either crystallisation, of the crystalline polymers, or with the glass transitionphase for the amorphous polymers.

A portable tip-sensitive temperature probe (Testo 720) was used to check that isothermal conditionswere maintained along the entire length of the melt barrel. This was done by inserting the probethrough the top of the barrel into the melt and recording the temperature as a function of position in the

barrel. The probe was also inserted into the second chamber, by removing the conical plug, and theheater cuff control was adjusted to give the same temperature as the barrel.




In the enhanced pressure experiments, a fixed ram speed was selected and the melt pushed throughthe die with the constriction initially fully open. At this point, the lower chamber pressure was close to

atmospheric. Both transducer readings were recorded simultaneously once the pressures had stabilised,i.e. the flow had reached a steady state. The constriction was then tightened, which resulted in theelevation of both the barrel and chamber pressures. Once the chamber had reached a predeterminedpressure, the pressures were again allowed to stabilise and the readings recorded. The chamber pressure

was incremented in this way, at intervals of approximately 6.8 MPa, up to a pressure of 61.2 MPa.When the run was completed, the process was repeated at another ram speed. In this way, the flow rate

was varied from 4.91Â10À9±2.45Â10À7 m3sÀ1.Fig. 2 contains a schematic diagram of the entry and exit regions for a melt flowing at a constant flow

rate Q. Also featured are the various pressures and pressure drops that are associated with flows througha capillary or an orifice. Assuming that the exit pressure drop is negligible, we have

ÁPo P2 À P1o (1)

and

ÁPc P2 ÀÁPo À P1cX (2)

In order to associate the pressure drops to a pressure level, we define mean values ~Po and ~Pc by meansof

~Po P2 P1oa2Y ~Pc P z ÀÁPo P1ca2X (3)

To determine the pressure drop in the capillary, which is required for the calculation of the shear

viscosity function, it is necessary to apply the so-called Bagley [20,21] correction. The main premise inthis correction is that the entry flows to the orifice and capillary are essentially the same at the sameflow rate and that they generate the same pressure drop. In our case, this condition will only hold

provided that the upstream pressures, as well as the flow rates are equal. This can always be achievedby adjusting the downstream pressure using the valve.

Fig. 2. Schematic diagram of the entry and exit flow regions around the orifice and capillary dies.




It is well established that, over ranges where the shear-rate dependence of the viscosity is essentiallypower-law in behaviour, the shear viscosity of the polymer melts, to a very good approximation,

increases exponentially with mean pressure. Then we can express the shear viscosity in the form of the

so-called Barus equation [22]:

0e sPY (4)

where 0 is the viscosity at atmospheric conditions and s is a pressure coefficient2. In what follows,we find that the extensional viscosity E has a similar exponential dependence, i.e.

E 0E e EPX (5)

To re-cast the raw pressure-drop/flow-rate results into the shear viscosity data, we use conventionalmethods, which accommodate the usual Weissenberg-Rabinowitsch correction. To re-cast the orifice

flow-rate/pressure-drop data, we apply the method pioneered by Cogswell [23] and extended by

Binding [24]. The analysis requires the shear viscosity function to be expressed as a power-law

k nÀ1Y (6)

where is the shear-rate and k and n are independent of . It results in another power-law relationshipfor the extensional viscosity E, which we write in the form

E l 4t À1Y (7)

where 4 is the extensional strain-rate and l and t are independent of 4.It could be argued that the results should be displayed as `reduced entrance-flow resistance' rather

than `extensional viscosity', since this is an accurate description of the data. However, we shall followour established custom of viewing the data as providing `a measure of resistance to extensional

deformation', which we describe by an extensional viscosity. We are, of course, fully aware of theliberties involved. In fact, the validity and limitations of the Cogswell±Binding approach for estimating

extensional viscosity have been well documented [23±25]. The most important point to make, perhaps,is that the analysis is based on considerations of viscous dissipation and so is limited to situations where

elasticity has a minor effect on the stress field. This is almost certainly the case for the flow of polymermelts through contractions at moderate flow rates.

Of interest is the dependence of and E on pressure and of significant interest in the present context

is the dependence of the Trouton ratio [26], Tr, on pressure, where

Tr

E 4

X (8)

To remove the ambiguity (the fact that the extensional viscosity is a function of stretch rate, while the

shear viscosity is a function of shear rate) in the choice of 4 and in Eq. (8), we make use of theproposal of Jones et al. [27], i.e.

Tr E 4

3p

X (9)

2Some authors refer to two coefficients, determined at either constant strain rate or at constant stress. This is a needless

complication since the two coefficients defined are simply related [28].




5. Experimental results

5.1. Low-density polyethylene (LDPE)

Tests on the LDPE were carried out at three temperatures: 170, 200 and 230 8C, which are within thenormal processing range. Examples of how the orifice and capillary pressure drops vary with meanpressure at 2008C are given in Figs. 3 and 4. The graphs are plotted on log/linear scales, and it is clearthat both the orifice and capillary pressure drops have an exponential dependence on pressure. The

orifice pressure drops represent around 25% of the uncorrected capillary data, a finding whichcompares favourably with that of Westover [2]. Westover found that the pressure drop associated with

entry flow for LDPE was between 25% and 50% of the total pressure loss for a capillary length-to-diameter ratio of 20.

The scatter in the raw data of Figs. 3 and 4 is relatively small, but it is inevitably more pronounced

for the orifice at the lowest flow rates. Here, the measured pressures represent less than 1% of the totaltransducer range. The apparent shear rates covered are relatively high for polymer melts with a

maximum level of 2.5Â103 sÀ1 being prescribed3. However, the maximum apparent shear rates here areno greater than those achieved by Laun [11] or Kadijk and van den Brule [15], who covered ranges upto 2Â103 and 3Â103 sÀ1, respectively.

Bearing in mind our earlier comments about power-law behaviour, it is not difficult to deduce (seeCouch and Binding [28]) that, for such behaviour, the dependencies on shear rate and pressure areexpected to be separable. The levels of the pressure coefficients recorded in Table 1 are moderate and

Fig. 3. Orifice pressure drop vs. mean pressure data with exponential curve fits for LDPE at 2008C.

3Needless to say, care was taken to restrict strain rates to below levels where fracture would be initiated.




exhibit no significant trends with temperature. It is worth mentioning that, when differences in the

definition of the pressure coefficients are taken into account, the coefficients obtained with our sheardata are in good agreement with those of Laun [13].

In Figs. 5 and 6, shear and extensional viscosities for the LDPE are displayed for differenttemperatures, at ambient pressure, and for different pressures at 2008C, respectively. shows the

Fig. 4. Bagley-corrected capillary pressure drop vs. hydrostatic pressure data for LDPE at 2008C.

Table 1

Shear and extensional pressure coefficients for the five polymer melt samples

Melt Temperature (8C) s (GPaÀ1) E (GPaÀ1)

LDPE 170 5.8 8.6

200 5.5 8.2

230 5.9 9.2

HDPE 150 5.6 14.1

170 5.9 15.7200 5.9 13.3

PP 190 7.6 19.3

200 6.4 17.9

230 6.9 15.5

PMMA 220 4.8 19.4

230 5.6 20.7

240 6.9 22.9

PS 180 8.5 29.2

200 7.4 32.7

230 8.4 31.1




Fig. 5. Shear and extensional viscosity vs. strain-rate curves for LDPE at 17,200 and 2308C, at ambient pressure.

Fig. 6. Shear and extensional viscosity vs. strain-rate curves for LDPE at 2008C for mean pressures of 0, 35 and 70 MPa.




expected shear-thinning behaviour, which is reasonably described by a power-law relationship. Theextensional viscosity is tension-thinning, and in this case power-law behaviour is, of course, imposed

on the data interpretation by the entry-flow analysis.In Fig. 7, the Trouton ratio results are displayed. This figure is important and allows us to write the

Trouton ratio in the separable form

Tr Tr0 4Y T e t P (10)

where t is sensibly independent of 4 and temperature. Values of t are given in Table 2. lt is evident

that the extensional viscosity E has a stronger dependence on pressure than the shear viscosity.

5.2. Polystyrene (PS)

Tests on PS were conducted at temperatures of 180, 200 and 2308C. The experimental data for the PSare interpreted in the same way as for the LDPE and the associated shear and extensional viscosities forPS at the three test temperatures, and ambient pressure, are plotted in Fig. 8. They indicate that the

polymer is shear-thinning and also tension-thinning, but only marginally so in the latter case. Althoughthere is some curvature in the shear data, the behaviour is still reasonably power-law, which is a pre-requisite in the present interpretation of the orifice data in terms of an extensional viscosity.

The variation of viscosity with mean pressure for the 2008C measurements is shown in Fig. 9.Clearly, the temperature and pressure dependence exhibited by E in Figs. 8 and 9 is far stronger than is

the case for the shear viscosity and is considerably greater than that of the LDPE

Fig. 7. Trouton ratio vs. stretch-rate curves for LDPE at 2008C for mean pressures of 0, 35 and 70 MPa.




The associated Trouton ratios are shown in Fig. 10. It is seen that Tr is a strongly-increasing functionof both shear rate and pressure, with an increase of nearly an order of magnitude with both variablesover the measured ranges. This is borne out in Table 2, where the relevant parameters can be seen to beconsiderably higher than for LDPE.

Table 2

Trouton ratio pressure coefficients for the five polymer melt samples

Melt Temperature (8C) t (GPaÀ1)

LDPE 170 2.7

200 2.7

230 3.4

HDPE 150 8.5

170 9.8

200 7.3

PP 190 11.7

200 11.5

230 8.5

PMMA 220 14.7

230 15.1

240 15.9

PS 180 20.8200 25.3

230 22.6

Fig. 8. Shear and extensional viscosity versus strain-rate curves for PS at 180, 200 and 2308C, at ambient pressure.




Fig. 9. Shear and extensional viscosity vs. strain-rate curves for PS at 2008C for mean pressures of 0, 35 and 70 MPa.

Fig. 10. Trouton ratio vs. strain-rate curves for PS at 2008C for mean pressures of 0, 35 and 70 MPa.




5.3. High-density polyethylene (HDPE)

Tests on HDPE were conducted at temperatures of 150, 170 and 2008C. The resulting pressurecoefficients, given in Table 1, are relatively low. The shear and extensional viscosities for HDPE at

2008C are shown in Fig. 11. The extensional viscosity curves are more limited in range in this case,because the orifice pressure drops were too low to measure at the lower flow rates. Although the shearviscosity again shows the expected shear-thinning behaviour, the extensional viscosity is weakly

tension-thickening.One consequence of the E behaviour given in Fig. 11 is that the associated Trouton ratios are

stronger functions of mean pressure than those of LDPE. This is confirmed in Fig. 12 and Table 2.

5.4. Polypropylene (PP)

The pressure coefficients, obtained for PP, displayed in Table 1, fall between those of LDPE and PS.

We mention in passing that the dependence on pressure that we observe for the shear viscosity is ingood agreement with the measurements of Kadijk and van den Brule [13].

The shear and extensional viscosity curves for PP at 2008C, given in Fig. 13, show similar strain-thinning tendencies to those of LDPE. However, the pressure dependence is stronger for PP, especiallyfor the extensional viscosity. Accordingly, the Trouton ratios for PP, given in Fig. 14, have an

appreciably stronger pressure-dependence than that of LDPE; but neither is nearly as high as thatof PS.

Fig. 11. Shear and extensional viscosity vs. strain-rate curves for HDPE at 2008C for mean pressures of 0, 35 and 70 MPa.




Fig. 12. Trouton ratio vs. strain-rate curves for HDPE at 2008C for mean pressures of 0, 35 and 70 MPa.

Fig. 13. Shear and extensional viscosity vs. strain-rate curves for PP at 2008C for mean pressures of 0, 35 ad 70 MPa.




Fig. 14. Trouton ratio vs. strain-rate curves for PP at 2008C for mean pressures of 0, 35 and 70 MPa.

Fig. 15. Shear and extensional viscosity vs. strain-rate curves for PMMA at 2308C for mean pressures for 0, 35 and 70 MPa.




5.5. Polymethyl methacrylate (PMMA)

The pressure coefficients for PMMA are second only, in magnitude, to those of PS. Fig. 15 shows theshear and extensional viscosity curves for the PMMA at 2308C, which appear to be very similar to

those of the PP, although the pressure-dependence of the extensional viscosity is stronger in the formercase. As a direct consequence of this, the Trouton ratios for the PMMA, shown in Fig. 16, exhibit agreater pressure-dependence than those of PP, but this is still considerably smaller than that for PS.

In general, the pressure coefficients of Table 1 and Table 2 display no significant trends withtemperature variation indicating that the pressure dependence is independent of temperature over theranges investigated. However, the level of uncertainty in the coefficients can be considerable,

particularly in the case of the Trouton ratio coefficients. However, this is not unexpected, as the rawdata is obtained by measuring relatively small differences in large pressure measurements. Further

considerations of the inter-relationship between temperature and pressure dependencies for the variouspolymers are addressed separately (Couch and Binding [28]).

6. Conclusions

The present paper is the first in a series that will address the important issue of the pressuredependence of rheological parameters in the case of commercially important non-Newtonian fluids. It

has concentrated on the extensional viscosity E and has relied on a popular method for estimating thisfunction. We are well aware of the difficulties of measuring extensional viscosity, even under

Fig. 16. Trouton ratio vs. strain-rate curves for PMMA at 2308C for mean pressures of 0, 35 and 70 MPa.




atmospheric conditions (see, for example, James and Walters [29]), but we believe that, at the veryleast, the contraction-flow technique provides a convenient measure of resistance to extensional

deformation and it is in that spirit that we communicate the results.

The pressure dependence of the melts considered in the present paper, as characterised by theTrouton ratio pressure coefficients in Table 2, is ordered PS > PMMA > PP > HDPE > LDPE.

This ordering must arise from the chemical and physical characteristics of the individual polymers,such as molecular weight, molecular weight distribution, density and compressibility, which will beexamined in more detail by Couch and Binding [28].

We conclude with the firm conviction that the basic findings of this paper are relevant and important.Indeed, the fact that the extensional viscosities have a substantial dependence on pressure, which is

greater than that of the shear viscosity on pressure, is particularly significant.

Acknowledgements

The authors wish to acknowledge significant support from ICI (Wilton, UK) and the DTI for itsearlier support through a LINK project, which led to the availability of our prototype capillaryrheometer. We also wish to express our gratitude to the EPSRC for its financial input to the current

research program. In addition we acknowledge the assistance of Mrs Christine Millichamp, Mr A.P.Palmer and Mr. J. Christiensen for their contributions to the experimental programme.

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Rhe Pressure Dependece

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