Stress relaxation in uniaxial extension

E L S E V I E R J. Non-Newtonian Fluid Mech., 67 (1996) 77-103

Jeuraalef l~ea-/~knvteaiaa

Fluid ~ d m l k s

Stress relaxation in uniaxial extension 1

N . V . O r r , T . S r i d h a r

Department of Chemical Engineering, Monash University, Clayton, Vic. 3168, Australia

Received 9 May 1996; in revised form 8 July 1996

Abstract

The elongation and subsequent relaxation of dilute polyisobutylene solutions in mixtures of polybutene and kerosene have been investigated using a filament stretching device first introduced by Tirtaatmadja and Sridhar [J. Rheol., 37 (1993) 1081-1102]. A novel experimental technique has been developed which enables the nature of the total stress measured by experiment to be investigated. A sudden decrease in stress is observed on cessation of the extensional flow. This is followed by a much slower relaxation. The results can be interpreted in terms of viscous and elastic stresses. An alternative explanation in terms of the dumbbell model is also explored.

The viscous stresses were observed to scale with strain rate at any given value of chain extension, whilst at steady state, the elastic stresses remained constant over a range of strain rates. Computations using a FENE-P model, with parameters evaluated from steady shear data, are shown to predict the growth of the elastic stress component. The same model seems to predict the relaxation of stresses reasonably well.

A new stretching technique has also been developed which allows excellent agreement between radially and axially computed strain rates, indicating homogeneous defomation of the fluid sample.

Keywords: Stress relaxation; Uniaxial extension

1. Introduction

T h e role o f ex tens iona l v iscos i ty m e a s u r e m e n t s in e luc ida t ing the b e h a v i o u r o f m a c r o -

mo lecu le s has been n o t e d in several reviews on this subjec t [1,2]. R e c e n t w o r k on the ex tens ion

o f di lu te a n d semi-di lu te p o l y m e r so lu t ions has h ighl ighted deficiencies in the abi l i ty o f cu r r en t

cons t i tu t ive e q u a t i o n s to accura te ly pred ic t t r ans ien t ex tens iona l p rope r t i e s [3], a n d in o u r

cu r r en t u n d e r s t a n d i n g o f the m e c h a n i s m s beh ind the t r ans ien t a n d s t eady-s t a t e responses o f

di lute p o l y m e r so lu t ions . Th is p a p e r is an a t t e m p t to use stress r e l axa t ion fo l lowing uniaxia l

Presented at the workshop on 'Unresolved Experimental Dilemmas in the Dynamics of Complex Fluids' at the Isaac Newton Institute, University of Cambridge, January 1996.

0377-0257/96/$15.00 Copyright © 1996 Published by Elsevier Science B.V. All rights reserved. PH S0377-0257(96)01487-5

78 N.V. Orr, T. Sridhar /J . Non-Newtonian Fluid Mech. 67 (1996) 77-103

extension to investigate the nature of the stresses generated during the unravelling of polymer molecules.

Larson [4] offered an excellent summary of available constitutive equations and hence provides a good starting point for this paper. The simplest constitutive equations are based on the well known Rouse and Zimm bead-spring models. Larson comments that when the frictional drag force that stretches the dumbbell exceeds the entropic restoring force of the spring, large extensions occur. At a critical strain rate 1/22, where 2 is the slowest relaxation time of the real polymer molecule, the frictional drag force and entropic force balance each other exactly. For strain rates below this value, the spring force dominates and the dumbbell collapses. Brownian forces in the polymer-solvent system oppose a complete collapse and hence the end-to-end distance of the dumbbell remains at a non-zero value, Rc.o. For strain rates above this value the drag force dominates, and as the entropic spring constant in the Rouse and Zimm models is independent of length, the drag force and spring force remain in the same relative proportion as the molecule extends. This leads to the well known problem of infinite stresses being predicted by this model at strain rates above the critical strain rate.

This model can be refined through the inclusion of finite extensibility, which essentially allows the spring constant to increase with extension, approaching infinity as the extension reaches some limiting value. However, Tirtaatmadja and Sridhar observed that the inclusion of finite extensibility offered only a slight improvement in the modelling ability of the constitutive equations used [3]. Further refinements to the Rouse model can be made by considering hydrodynamic interaction and internal viscosity.

In the Rouse model, it is assumed that the solvent veolcity near one bead is independent of the effect of neighbouring beads. This can be envisaged by imagining the solvent flowing through a swarm of molecules. If the distance between the molecules is large enough, the solvent moves freely between the molecules and is referred to as being "freely draining". Hydrodynamic interaction can also result in shear-thickening. This is caused by individual portions of the chain influencing the solvent flow around each other, resulting in an increase in the drag force. It is this increase in drag force which results in the stress behaving non-linearly with extension.

A further refinement to the Rouse model assumes that the polymer chains are unable to resist the deformation imposed by the associated fluid element. Hence, if one assumes that frictional barriers to short-range distortions make the polymer chain somewhat rigid and resistant to rapid deformation, the concept of internal viscosity emerges. In terms of modelling, internal viscosity can be represented in the bead-spring model through addition of a linear dashpot connected in parallel with the bead connector. Thus one would expect the inclusion of internal viscosity to result in an additional stress which is strain-rate dependant and purely viscous in nature.

Manke and Williams [5,6] have investigated the properties of the spring-dashpot model in detail. Extending their analysis to multi-bead spring models with internal viscosity (IV), the authors noted that as the magnitude of IV approached very large values, the behaviour of the bead-spring model approached that of the freely jointed bead-rod model. Manke and Williams results suggest that for moderate values of IV, the physical system can be envisaged as a combination of flexible and rigid molecules.

The aforementioned theoretical modifications to the simple dumbbell theory can all result in polymer stresses which grow non-linearly with extension. However, in all cases but internal viscosity, the factors influencing the non-linear stress growth with extension are not strain rate

N.V. Orr, T. Sridhar / J. Non-Newtonian Fluid Mech. 67 (1996) 77-103 79

dependant. Hence evidence of the existence of strain rate dependant stresses would help to differentiate between the contributions of these theories and also their origin.

Liang and Mackay [7] observed a stress jump at the cessation of steady shear for a semi-dilute system of semi-rigid rod molecules. They comment that the observed stress jumps were above and beyond the stress jump associated with the solvent contribution. The authors suggest that up to 50% of the original polymer contribution to the stress was lost upon cessation. The initial fast decay was assumed to be purely viscous dissipation. Smyth et al [8] compared the birefringence of the same solution as used by Liang and Mackay and found that when the birefringence was plotted against total polymeric stress, the stress-optic law did not hold. However, when only the "elastic" stress was considered, the stress-optic law was recovered.

Hua and Schieber [9] attempted to model the experimental data of Liang and Mackay [7]. They found that a finitely extensible dumbbell model including IV was able to predict both the elastic and viscous components of the measured stress. However, the ability of this model to predict the viscous component of the total stress tells us nothing about the molecular configura- tions and behaviour which give rise to this effect in the first place, therefore we need to examine other theories which attempt to explain the discrepancies between experimental data and theoretical predictions.

The observation of James and Saringer [10] that the stresses obtained in sink flow were greatly underestimated by the Rouse model led King and James [11] to propose that slip existed between the polymer molecule and solvent flow. They suggest that such an effect could be produced by an extending polymer molecule prematurely freezing in a partially extended configuration. They attributed this "freezing" to the effect of self-entanglements or knots". The authors postulate that extending molecules may increasingly resist further extension, and hence the stresses will approach that of rigid molecules.

Ryskin [12] offered a different interpretation of James and Saringer's experimental results. He suggests that viscous drag force on the molecule increases faster than linearly with extension, and is maximum at the middle of the polymer chain. The author further suggests that the resulting fast increase in the drag force causes the polymer molecule to unravel affinely with the solvent flow until the chain is nearly fully streteched and as the drag force is maximum in the middel of the chain, the polymer molecule may actually stretch in the middle and remain coiled at the ends. Ryskin referred to this model as the "yo-yo" model.

The author argues that the central portion of the molecule acts hydrodynamically as a rigid-rod, hence generating a large purely viscous stress. Ryskin further comments that unlike the simple dumbbell model which assumes that the external mechanical energy spent on working against the stress transforms entirely into the elastic energy of the stretched chain, the "yo-yo" model assumes that the stored elastic energy is insignificant and the stress is entirely dissipative.

Rallison and Hinch [13] attempted to explain the inability of simple constitutive equations to describe experimental observations in extensional flow by performing numerical simulations on a multi-bead-rod model as an approximation to a real polymer molecule. In this model, the beads were considered to be freely jointed, and Brownian motion was neglected as the flow strength was strong. The authors observed that up to a strain of approximately 2, the chain appeared to stretch affinely with the flow and viscous dissipation was relatively low. However, after this small strain, the chain stretched more slowly and "slip" between the chain and the solvent flow resulted in an additional stress which was dissipative in nature. Rallison and Hinch

80 N.V. Orr, T. Sridhar / J. Non-Newtonian Fluid Mech. 67 (1996) 77-103

attribute this to the inability of the chain to conform to the fluid motion and to "kinks" in the chain (or back loops) which are prevented from unfolding by the compressional part of the flow in the direction perpendicular to the principal stretching direction. The authors comment that as stretching continued, the chain stretched at only 35% of the flow rate. Hence large viscous stresses were developed.

Larson [14] extended the simulations of Rallison and Hinch by developing equations describing "kink-dynamics" i.e. how a chain evolves from a highly kinked state to fully stretched. He suggests that as the number of kinks decreases, the ability of the chain to respond affinely to the motion of the solvent is reduced. The stress produced by these kinked molecules was closer to that produced by rigid molecules than for those which deform affinely. However, the author notes that the average end-to-end distance, R of the kinked chain grows nearly affinely except when the chain is close to full extension. Therefore, despite the nearly affine motion of the ends of the kinked molecule, the mismatch at most points along the chain contour between the solvent velocity and that of the chain produces a stress comparable to an inextensible strand of length R. Due to the nearly affine motion of the ends of the kinked molecule, the bead-spring model should give a reasonable description of the end-to-end distance up to the point where significant departure from affine motion occurs.

Hinch [15] elaborates on the arguments proposed by Rallison and Hinch [13] and Larson [14]. Hinch observed that as the flow strength was increased by a factor of 5, so did the stress. This result is not what one would expect from the elastic dumbbell, but is in line with dissipative stress produced by rigid molecules. The author suggests that as the strain increases, the deformable chain become more rigid in behaviour as it unfolds. Also until the chain is nearly fully stretched, each part of the chain evolves in a similar way, suggesting that the stress should scale with molecular weight. Rallison [16] has investigated the behaviour of a bead-spring model in a planar extensional flow using a Brownian dynamics simulation. The study shows that a stress, apparently viscous in nature, develops along with the elastic stress and is of comparable magnitude. Rallison shows that the magnitude of the dissipative stress depends on experimental timescales, and suggests that the viscous stress is a pragrnetic way of including the effects of the shortest relaxation times. We return to this issue later in the paper.

Another possible reason why a polymer molecule fails to conform affinely with the solvent flow can be inferred from the results of James and Sridhar [17]. Based on calculations using the theory of Batchelor [18], the authors observed that dilute polymer solutions reach a steady state in extensional flow at much lower values of polymer extension than one would expect from the total length of the molecule. The authors propose that self-entanglements may have caused the discrepancy between actual and maximum polymer extension.

Brochard and De Gennes [19] comment that knots are more likely to occur in poor solvents than in good ones, as the chain is more compact in a poor solvent due to negligible excluded volume effects. The authors derived an expression for the stress contribution of single-entangled polymer which included a component that was strain rate dependant. Hence, the authors predict that the presence of knots or self-entanglements results in a viscous, dissipative stress term. Vologodskii et al. [20] studied the probability of knot formation using a multi-bead-rod chain undergoing a random walk. The authors observed that for molecules with small aspect ratio, the probability of knot formation was extremely small. However, simulations approaching chains with large aspect ratios showed that the probability of knot formation was good. The authors


predicted a trend in which the probability of knot formation increases linearly with molecule length.

Although the above discussions differ in the exact mechanisms which produce the additional viscous term, they all have one major point in c o m m o n - t h a t is that the viscous stresses are produced by behaviour approaching that of rigid, inextensible objects, and that the total stresses are predominantly dissipative.

The aim of this paper is to experimentally probe the nature of the stresses by investigating the relaxation of stresses after extensional deformation and to enquire if the experiments support the existence of a significant viscous stress. By decomposing the total stress into a viscous and elastic portion, it is hoped that the effects of the various physical mechanisms can be isolated, hence increasing our ability to accurately predict the response of polymer solutions in extensional flow.

2. Experimental technique

The fluid preparation, composition and shear prope.rties have been previously described [21] and hence will not be repeated in this paper. The PIB Boger fluids used in these experiments are labelled Fluid A and Fluid B and have molecular weights of 1.2 x 106 and 2.4 x 106 and viscosities at 21.5°C of 19 Pa.s and 34.2 Pa.s respectively. Unless otherwise stated, all experiments were conducted at 21.5°C.

Two different drive systems were used to stretch the polymer samples. The first extensional rheometer was identical to that described by Tirtaatmadja and Sridhar [21], and consisted of two platforms mechanically coupled to a single d.c servo motor. The correct velocity profile was achieved by using an arbitrary waveform generator to supply an exponentially increasing voltage to the servo motor.

The second extensional rheometer used a linear drive unit complete with servo controller. This unit consisted of two independent platforms located on a common shaft, and each platform is controlled individually. Movement was achieved by activating electromagnets within each platform, which in turn interacted with permanenet magnets embedded in the common shaft. The servo controller consisted of a PID control system, which allowed each platform to move to a preset position over a given time interval. The desired separation of each platform with time is entered in array form at 10 ms intervals. The error between the desired position and actual position was measured every 2 ms, and servo power was adjusted accordingly.

In each of the two drive systems, a Cambridge Technology 400A force transducer is attached to one of the platforms whilst a fixture is attached to the other. Aluminium disks of various diameter (end plates) can be attached to the force transducer and an identically sized brass rod is attached to the fixture. The manufacturers quote a response time of 1 ms for the 400A unit and a lower limit on the measurable force of 2.5 x 10 -7 N.

An optical diameter-measuring device (Zumbach ODAC) is used to measure the diameter of the filament as it is stretched. A beam of light is transmitted and the object to be measured is placed in its path. The amount of light detected by the device after traversing the object determines the diameter. Tirtaatmadja and Sridhar [21] report that the Zumbach is accurate to within + 2°/'0 at a lower limit of 0.05 m m and has a resolution of 0.005 mm. In the situation where both platforms are moving, the Zumbach is positioned at the mid-point of the filament,


and hence monitors the filament mid-point diameter as the filament is stretched. In the situation where only one platform is moving (say platform holding force transducer is stationary), the Zumbach is offset from the mid-point of the filament to ensure that the diameter measurements are not affected by conditions at the end plates.

The voltage signals from the force transducer and diameter measuring device are collected into a PC using Labtech for Windows software and sampled at 10 ms intervals via an analog-to-digital card (ADC). Several signal conditioners are incorporated into the ADC to improve the quality of the data collected. The cut-off frequency of the analog filters was increased in order to capture fast transients. It was found that a cut-off frequency of 40 Hz gave a good compromise between signal quality and noticeable lag due to filtering, and was confirmed by comparing the processed and un-processed signals on a digital oscilloscope.

During any experiment the fluid sample was loaded onto the brass rod, which was then mated with the aluminium end plate attached to the force transducer to obtain a cylindrically shaped specimen whose diameter was identical to that of the disks. The initial sample length varied from 1 to 1.5 mm with an initial filament diameter of 3-12 mm. Tirtaatmadja and Sridhar [21] have shown that for ideal extensional flow the instantaneous

length of specimen L ( t ) must increase with t as

L ( t ) = Lo exp(~t), (1)

where Lo is the initial filament length and ~ is the imposed strain rate on the filament. Hence, the filament diameter must decrease from an initial value Do by

D ( t ) = Do e x p ( - l ~ t ) (2)

In a real experiment, the fluid behaviour deviated from Eqs. (1) and (2) due to end effects caused by fluid adhering to the end plates. The method adoped by Tirtaatmadja and Sridhar to compensate for these end effects is to adjust the velocity of the drive units by trial and error until constant extension rates based on filament mid-point diameter are achieved for the entire duration of the experiment [21]. This was achieved by modifying Eq. (1) such that

for t < tl V( t ) = k Lo exp(kt), (3)

for t > tl V(t) = k Lo exp(rkt) , (4)

where V(t) is the drive velocity, k is generally equal to half the strain rate, and r and t I are adjustable parameters.

Due to the ability of the linear drive unit to rapidly effect position movements, a second technique was developed. With this technique, the fluid sample was first subjected to a small step strain, then sample length redefined before a constant strain rate was imposed such that

for t = 0 - L ( t ) = Lo, (5)

for t = 0 + L ( t ) = Ls, (6)

for t > 0 L ( t ) = Ls exp(~t), (7)

where Ls is the length of step. The resulting filament is uniform and cylindrical along the majority of its legth, although initial stresses are non-zero due to the rapid change in length. In order to


compare results obtained from these different stretching techniques, Eq. (1) can be rearranged to give an equivalent experimental time required for an unstepped sample to increase in length from Lo to Ls such that

t= 1/~ ln(LJLo). (8)

The results obtained from the step-strain experiments are then offset by the time given in Eq. (8). In order to determine the magnitude of the dissipative (viscous) and elastic components of the

polymer contribution to the total stress, a novel experimental technique was developed which enabled these two components to be separated. At any desired time during a typical stress growth experiment the drive velocity was set to zero (the drive can come to a halt in about 50 ms, dependant on the velocity at the moment prior to stopping). The filament diameter remained constant and hence allowed the stress relaxation to be monitored. It was observed that rapid changes imposed on the force transducer during the deceleration process caused the initial stress relaxation data to oscillate. Hence the residual stress relaxation was fitted with the sum of two exponentials,

a = A e x p ( - t/a) + B e x p ( - t/b). (9)

Eq. (9) was used to extrapolate the stress relaxation back to the time at which the drive came to a halt, and the corresponding value was labelled as elastic stress. The difference between the value of stress obtained at the end of the stretching experiment (total stress) and the elastic stress was labelled as viscous stress. Details of this procedure are given in Fig. 1. By repeating this procedure for different final extensions (or stretching time), a discrete picture of the extensional response of each stress component was formed as shown in Fig. 1. Also demonstrated in this figure is the repeatability of the stress growth experiments.

3. Results

In order to determine the effects of the non-homogeneous extensional behaviour on the large-strain behaviour of the test solutions, a sample of fluid B was extended according to Eqs. (3)-(7). The step-strain experiments were conducted with and without a pause after the initial step (Eq. (6)) to determine the effects of prestressing the fluid, and each step-strain experiment was offset according to Eq. (8) in order to compare the two different methods. The effect of different aspect ratios prior to uniaxial extension were also investigated (Fig. 2).

For the experiments conducted according to Eq. (3) and (4), the stress growth data shown in Fig. 2 exhibits negligible differences with respect to Lo/Do, with each experiment also passing through a Trouton ratio of three before the onset of strain hardening. For the step-strain experiments, the appropriate definition of the initial aspect ratio is Ls/Ds, where the subscript refers to the length and diameter after the initial step. The experiments conducted according to Eqs. (5), (6) and (7) show similar stress growth behaviour to the previous method, though the initial value of stress is dependant on the length of pause after the step-strain. The stress growth is seen to eventually coincide with the data obtained without a step. The time required for this coincidence is short if a pause is included after the initial step such that the stress at the commencement of stretching is small. As the aspect ratio at the start of uniaxial extension varies


for each method by an order of magnitude, the small differences observed in the stress growth curves indicate that the dependence of stress growth on the aspect ratios used in these experiments is small. Note that the steady states obtained are identical for all four experiments.

The diameter and force profiles obtained from the experiment conducted according to Eqs. (5), (6) and (7) with no pause are displayed in Fig. 3. Clearly shown in this figure is the near constant exponential decrease in diameter with time, indicating that the experiment was conducted under near-ideal conditions of constant strain rate based on both length and

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/

Total Stress /

Elastic Stress

0 0.5 1.5 2

time (sec) Fig. 1. Total stress growth for fluid A at a strain rate of 4 s- 1. Experiment halted at different times. Inset shows the determination of viscous and elastic stresses.

N.V. Orr, T. Sridhar / J. Non-Newtonian Fluid Mech. 67 (1996) 77-103

1E+04

85

O

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e I -

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L s l D s = 7.6 - Stepped 8mm then Single E x p (2 sec pause)

1 1.5

time (sec)

Fig. 2. Total stress growth for fluid B at a strain rate of 3 s - t . Fluid sample stretched using two different velocity profiles and initial conditions.

diameter. The strain rates evaluated from Eq. (1) and (2) vary by less than 10%, clearly indicating that the deformation was uniform. Fig. 3 also clearly shows a maximum and subsequent decrease in the force curve with time. This experiment lasted about 0.35 after the maximum in the force curve was reached. During this period of decreasing force, the stress increases by at most 50%. During a similar time period prior to the maximum, the stress increases by a factor of approximately 20.

Initial relaxation experiments were conducted on the Newtonian solvent HYVIS30 (low molecular weight polybutene) to determine the nature of viscous stress relaxation after stretching


on the extensional rheometer. At the conclusion of a normal stretching experiment of a given duration, the rheometer was instructed to stop, and the force, filament diameter and motor tachometer continuously recorded. It ws observed that the drive unit required approximately 40 ms to come to rest (Fig. 4). During this deceleration period both the strain rate and force decrease linearly, and it can be seen that the force transducer responds immediately to changes in acceleration generated by the d.c. servo motor feedback control system. Fig. 4 also

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0.7

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Fig. 3. Force and diameter data during and after extensional flow for fluid B at a strain rate of 3 s-l . Aspect ratio before extension is 7.6 (no pause). Note the rapid decay in force and constant diameter after cessation of extensional flow.

N.V. Orr, T. Sridhar / J. Non-Newtonian Fluid Mech. 67 (1996) 77-103

0.2 ~ 1

8 7

0.18

0.16

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0.08

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Fig. 4. Force relaxation and diameter for newtonian solvent HYVIS30 after an extensional strain rate of 3.2 s -~. Also shown in the rheometer velocity during deceleration.

demonstrates that the filament diameter remains constant during the relaxation process, and photographs of the filament (not shown) confirmed that over a period of several seconds, the diameter was constant and uniform along the length of filament.

In contrast, for the polymer solutions, the relaxation stress was observed to consist of two components; a rapid decay and a residual stress which exhibited a slower exponential decay. Examples of this type of stress relaxation are displayed in Fig. 1 and 3. In order to assess the effect of the finite time required to decelerate the rheometer on the residual stresses, a sample of fluid B was stretched at a rate of 2 s-~ for an arbitrary length of time. The drive was then decelerated at different rates whilst the force transducer was kept stationary to prevent any

88 N. I1. Orr, T. Sridhar /J . Non-Newtonian Fluid Mech. 67 (1996) 77-103

oscillations due to the imposed deceleration. As a result, the time required to decelerate the rheometer varied from 50 to 610 ms, where 50 ms was the maximum achievable deceleration under these circumstances. These results are shown in Fig. 5a, and are offset such that the time at which the drive units actually came to rest coincide for each experiment.

It was found that the initial rate of decay was proportional to the rate at which the rheometer was decelerated, and that the residual stresses for all but the slowest deceleration showed an

80000

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60000

50000

40000

30000

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--o- Ramping 50ms

-~- Ramping 100ms

-a - Ramping 180ms

-o - Ramping 390ms

--x- Ramping 610 ms ! J

0 0.2 0.4 0.6 0.S 1

t ime (sec)

Fig. 5(a). Stress relaxation following extensional flow for fluid B at a strain rate of 2 s - ] and temperature of 24.5°C. Force transducer kept stationary during experiments and time to completely stop drive unit varied. Data are offset such that times at which drive unit came to complete rest coincide for each experiment.


extremely weak dependence on the time required to stop the rheometer. The initial rapid stress relaxation is termed "viscous dissipation". It is recognised that the experiment is unable to distinguish between an instantaneous viscous stress relaxation and a fast elastic stress relaxation of duration 50 ms or less.

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70000

60000

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20000

10000

--o- Ramping 50ms - Offset 10 ms

- - . - Ramping 100ms - Offset 20 ms

- - - - Ramping 180ms - Offset 40 ms

--o- Ramping 390ms - Offset 80 ms

--x-- Ramping 610 ms - Offset 130 ms

0 ~

0 0.2 0.4 0.6 0.8 1

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Fig. 5(b). Stress relaxation following extensional flow for fluid B at a strain rate of 2 s-1 and temperature of 24.5°C. Force transducer kept stationary during experiments and time to completely stop drive unit varied. Data are offset such that times at which critical strain rate is reached coincide for each experiment.


It can be seen from Fig. 5a that the residual elastic stress relaxation for the slowest deceleration (corresponding to a ramp time of 610 ms) appears to begin relaxing before the drive system comes to complete rest. During the deceleration period, the strain rate decreases linearly. During a portion of this period, the polymer molecules are exposed to strain rates below the critical strain rate. If it is assumed that a polymer molecule would not start relaxing until the imposed strain rate falls below a critical value ec, the differences in the residual stresses may reflect the length of time during the deceleration period that the strain rate is below ~c. The critical strain rate can be determined using the longest relaxation time of the polymer solution, where g~ is equal to 1/2Z This relaxation time can be found from low frequency shear data. However, the slowest relaxation time of the polymer molecule after extensional flow should give an identical value to that obtained from shear data. Such data have been obtained for fluid A and it was observed that the single mode relaxation time obtained from shear data (refer to Table 2) and extensional relaxation experiments (from Eq. (9)) were identical. As the available shear data for fluid B does not extend to low enough frequencies for an accurate estimate of the relaxation time [22], the longest relaxation time obtained from Eq. (9) was used to calculate the critical strain rate for fluid B. Thus ~ corresponds to a value of 0.4 s-I for fluid B at 24.5°C. The stress relaxation curves depicted in Fig. 5a were then sifted such that the times at which the imposed strain rate dropped below the critical strain rate coincided for each experiment (Fig. 5b).

Fig. 5b clearly shows a collapse of the residual elastic stress decay for all ramping times, which implies that the residual elastic stresses do not start to relax until the linearly decreasing strain rate passes through the critical strain rate. For the faster deceleration curves, this proportion of the total ramping time is small (one sampling period), and hence it can be assumed that no elastic stress relaxation takes place till the rheometer comes to complete rest.

By varying the time at which the rheometer is stopped (Fig. 1) one can obtain viscous and elastic stress at any value of the total stress. Results from such experiments are shown in the following figures. Fig. 6a and 6b show the transient total, viscous and elastic stress growth data for fluid B obtained at strain rates of 2 and 3 s-1 respectively. Clearly displayed in these figures are the asymptotic behaviour of both the elastic and viscous stress components as steady state is approached. The stress behaviour as steady state is reached is displayed in more detail in the inset on these figures. Oscillations in the values of the stress data collected near the end of the experiments are due to noise generated by movement of the force transducer and filament, and are observable due to the lower degree of signal processing adopted in this work. In Figs. 6a and 6b, the last two elastic stress data points shown can be thought to correspond to a steady state situation, Hence, it is of interest to examine the stress relaxation for these two cases. Fig. 7 shows that stress relaxation under these conditions are identical suggesting that the polymer molecules are in a similar configuration at the start of relaxation. As it has been previously shown that the elastic stress does not appreciably decay until the drive unit has come close to complete rest, the relaxation curves have been truncated before this time.

Evident in Fig. 6a and 6b are the similarities in the values of elastic stress obtained at steady state. In contrast, the values of viscous stress differ markedly, and are in proportion to the strain rate. For example, in Fig. 6a the elastic stress is 60458 Pa with an associated viscous stress at a strain rate of 2.1 s-~ of 94330 Pa. Comparing these results to those obtained from Fig. 6b, at a strain rate of 3 s -1 the elastic stress is 57719 Pa with an associated viscous stress of 156975 Pa. The differences in the elastic stress values are small for the two strain rates, however the

N.V. Orr, T. Sridhar / J. Non-Newtonian Fluid Mech. 67 (1996)77-103 91

viscous stresses show a large dependence on strain rate. Also of interest is the highly non-linear relationship between the transient elastic and viscous stress growth observed in these figures. It is evident that the magnitude of the viscous stress increases much more rapidly with time when compared to the elastic stress in each of the two figures.

Differences in the behaviour of the viscous and elastic stresses with strain rate are readily observed by comparing the stress relaxation from steady state for different strain rates. This is demonstrated for fluid B in Fig. 8. Note the dependence of the total stress on strain rate. Fig.

1E+06

1E+04 A a n v

U)

1E+06

1 E . O . ~"mm ~ ~o~-

5 r • ~ 1E+04

1E+05 I E + 0 3

1E+03

1E+02

1 E+O 1

2.2 2.4 2.6 2.8 3

n Total Stress Data

• Elastic Stress

• Viscous Stress - Solvent Stress

0 0.5 1.5 2 2.5 3

t i m e ( s e c )

Fig. 6(a). Total stress growth for fluid B at a strain rate of 2.1 s- I. Viscous and elastic stress components obtained from stress relaxation experiments are also shown. The solvent contribution has been subtracted from the viscous stress. Inset shows detailed view of stress growth near steady state.


8 also clearly shows that the residual elastic stresses are independent of strain rate. These values are also summarised in Table I. This result indicates that at steady state, the contour length of the macromolecule is identical for each strain rate. It is interesting to compare this result with the predictions of the FENE-P bead-spring model. At steady state, the FENE-P model predicts that the contour length of the elastic dumbbell increases with strain rate, although the increase

A

o. v

(0 .o

u)

1E+06

1E+05

1E+04

1E+03

1E+02

1E+01

1E+06

1E+05 I

1E+04 /

1E+03 |

&

~ , ~ 0 ~ - ~ ~ ' . t

1.5 1.6 1.7 1.8 1.9

I I o Tota l St ress Data

, ) Elastic Stress • Viscous Stress - So lven t St ress

/ .

/ J

F 4'

0 0.5 1 1.5 2

t i m e (sec)

Fig. 6(b). Total stress growth for fluid B at a strain rate of 3 s-I. Viscous and elastic stress components obtained from stress relaxation experiments are also shown. The solvent contribution has been subtracted from the viscous stress. Inset shows detailed view of stress growth near steady state.

N.V. Orr, T. Sridhar /J. Non-Newtonian Fluid Mech. 67 (1996) 77-103 93

may be very small. It should also be noted that the FENE-P model does predict that at steady state the polymer contribution to the total stress scales with strain rate. This arises due to the F E N E spring constant, which varies linearly with the strain rate. However, the results shown in Fig. 6a and b suggest that the differences in the total stress observed experimentally are due to the dependence of an extra viscous stress component on strain rate (refer also to Table 1). Hence, we propose that the total stress may be written in the form

O" T = O" E + O" V. (10)

100000

80000

A

o. v m m 60000 ,~

U) c 0 (w x m

m

40000

20000

Relaxation at Start of Steady State

--a- Relaxation During Steady State

d

0 0.2 0.4 0.6 0.8 1

t ime (sec)

Fig. 7. Steady state stress relaxation for fluid B after extension at a strain rate of 3 s- ]. Curves correspond to the last two relaxation experiments in Fig. 6b.

94 N.V. Orr, T. Sridhar ,/J. Non-Newtonian Fluid Mech. 67 (1996) 77-103

3E+05

2E+05

A

o. --.- 1E+05 W

0E+00

Stra in Rate =

4/s

Strain Rate =

3Is

Stra in Rate =

1.6Is

m m m , , m

- ~ - S t r a i n Rate = 3 I s

- - . - S t r a i n Rate [ ] 1 . 6 / s

- e - Strain Rate = 4/s

-1E+05 0 0.25 0.5 0.75

t ime (sec)

Fig. 8. Steady state stress relaxation for fluid B after extension at strain rates of 1.6,3 and 4 s-1. Note differences in initial total stress with strain rate.

Due to the strain rate dependence of trv, at higher strain rates one would expect the total stress to be predominantly viscous, which is in line with the observations of Hinch [15] and Larson [14] and is also supported by the trend shown in Table 1.

Plotting the viscous stresses observed for fluid B at steady state against strain rate demonstrates a linear relationship (Fig. 9). By interpolation, viscous stresses corresponding to different constant values of elastic stress can be obtained and are also displayed in Fig. 9. The results shown in this figure demonstrate that a linear relationship is observed between the viscous stress and strain rate at any constant ~value of elastic stress, and hence chain extension. In order to calculate the relationship between viscous stress and elastic stress, the viscous stresses were first

N.V. Orr, T. Sridhar /J. Non-Newtonian Fluid Mech. 67 (1996) 77-103 95

converted to extensional viscosities. This enabled a range of viscous stress values corresponding to different strain rates to be plotted against elastic stress as shown in Fig. 10. Clearly shown in Fig. 10 is the non-linear dependence of viscous stress on elastic stress discussed previously. Now that a relationship between the viscous and elastic stress has been established, Eq. (10) can be rewritten as

O" T = O" E -[- k ~ { O" E } at, (11)

where ~ = 1.65 f rom Fig. 10. This results, al though not shown, also holds true for fluid A and can be compared with the analysis of Liang and Mackay, whose predictions suggest that for semi-rigid rods in semi-dilute solution, ~ = 2 [7].

For the purposes of modelling the extensional flow of dilute polymer solutions, it would be useful to compare the predictions of a theoretical model with experimental stress growth data. Referring to Larson [14], in terms o f the "kink model", the average end-to-end distance, R of a kinked chain grows nearly affinely except when the chain is near to full extension. Hence, the bead-spring model should given a reasonable description of the end-to-end distance up to the point where significant departure from affine mot ion occurs. Figs. 1 l a and b show the transient total and elastic stress growth data for fluid A obtained at strain rates of 2 and 3 s - respectively. Also displayed on these figures are FENE-P model predictions, where the govern- ing equations and parameters have been obtained from Tir taatmadja and Sridhar [3] and are displayed in Table 2. It should be noted that the parameters used in the FENE-P model have been obtained by fitting the model to dynamic and steady shear data, and hence provide an independent means of predicting extensional stress growth. As with Figs. 6a and b, the asymptotic behaviour o f the total and elastic stress is observed. Fluid A appeared to be more susceptible to noise near the end of the experiments (possibly due to lower viscosity, hence more filament movement) , therefore the stress behaviour as steady state is approached is shown in greater detail in the inset in these figures.

Table 1 Steady state values of stress components for fluids A and B at all strain rates

Strain rate Trouton ratio Total stress Elastic stress Viscous stress Solvent (s -1) (Pa) (Pa) (Pa) contribution (Pa)

Fluid A 2.1 3467 140000 61409 78470 121 3 3509 200000 63079 136750 171 3.5 3359 225000 67221 157578 201 4 3249 250000 64586 185183 231

Fluid B 1.6 2219 120000 56837 63000 162 2.1 2194 155000 60458 94330 214 3 2108 215000 57719 156975 306 3.5 2000 240000 57530 182110 360 4 1961 270000 58174 211826 413

96 N.I,'. Orr, T. Sridhar / J. Non-Newtonian Fluid Mech. 67 (1996) 77-103

2.5E+05

2.0E+05

"~ 1.5E+05 G.

W 2

o Io N 1.0S~S

5 . 0 E ~ 4

0.0E+00

o Elastic Stress = 58,000 Pa (Steady State)

o Elastic Stress : 27,000 Pa

a Elastic Stress : 20,000 Pa

I I I I

0 1 2 3 4 5

Stra in Rate (1Is)

Fig. 9. Viscous stress for fluid B versus s t ra in ra te at three cons tan t values of elastic stress. No te tha t an elastic stress of 58 000 Pa cor responds to s teady state.

Clearly shown in Fig. 1 la, the FENE-P model appears to predict the elastic stress growth and subsequent steady state remarkably well, especially considering that the model parameters have been obtained from shear flow data and are independent of the extensional results. In addition to the above FENE-P calculations, the model was modified according to Eq. (11), with the elastic stress being calculated by the FENE-P model as above. The modified FENE-P calculations are also displayed in Fig. 1 la. Clearly demonstrated in this figure is the improvement in the prediction of the total stress. This result suggests that during transient stress growth and subsequent steady state, the viscous stress is always dependant on the level of elastic stress and


highlights the need for an additional viscsous dissipation term in the FENE-P model. At a higher strain rate (Fig. 1 lb), the model appears to over-predict the elastic stress growth and also shows an abrupt approach to steady state. Al though not shown in this paper, FENE-P predictions were also obtained for strain rates of 1.6, 3.5 and 4 s - i . It was observed that as the strain rate was increased, the predictive ability of the FENE-P model decreased.

Also of interest for modelling purposes is the ability of the FENE-P model to predict the stress relaxation at different levels of polymer extension. This is investigated in Fig. 12a and b.

o

ILl W -s 0 0 .m

1E+05

1E+04

1 E+03

1E+02

oj

f/ ! 1 E+O t

1 E+02 1 E+03 1E+04 1 E+05

Elast ic St ress (Pa)

Fig. 10. Viscous stress plotted as extensional viscosity for fluid B over a range of strain rates versus elastic stress. Treadline shown on plot has slope of 1.65.


Table 2 Single and mult i-mode F E N E - P parameters obtained from Ref. [21] (shifted to 21.5°C).

Mode qK(Pa.s) 2K(S) b

1 1.53 3.8 5.5 2 2.32 1.017 6000 3 2.29 0.151 1700 4 1.673 0.0135 1000 Single mode 7.81 1.1 - Solvent 11.22 - -

The model predictions for relaxation behaviour were performed over three distinct regions: (i) a region of constant strain rate stress growth, (ii) a region of linearly decreasing strain rate (corresponding to the period during which the drive is coming to a halt) and (iii) a region of zero strain rate stress relaxation. It should be noted that the model parameters used for relaxation were identical to those used for stress growth calculations. As with the previous results, for the lower strain rate (Fig. 12a) the ability of the model to predict the stress relaxation is extremely good over a range of polymer extension. However, note that the FENE-P model only captures the trend at the later stages of the relaxation. The model does not predict the initial rapid drop in stress, especially at small times (or alternatively, at chain extension below steady state). A detailed view of experimental data versus FENE-P predictions corresponding to the smallest value of extension is shown in the inset.

It could well be argued that the total stress data displayed in the above figures may as well be represented by the usual FENE-P model with additional parameters which relax rapidly, and hence on the time scale of the experiment appear to be a viscous dissipation [16,23]. Such fast relaxation modes may not be apparent in dynamic stress experiments. We examine now the advantages and disadvantages of such a representation.

The FENE-P model predicts a steady state extensional viscosity which is essentially independent of strain rate. As previously discussed, it obtains this by a combination of a nearly constant chain extension and a spring constant which varies linearly with strain rate. Such a representation would indicate that at the total stress plateau shown in Figs. 6 and 11, the chain is nearly fully extended. In order for the FENE-P model to capture the magnitude of the steady state stresses, it would be necessary to alter the constant 'b' in the model to much larger values i.e. make the spring more extendable and hence more Hookean. Unfortunately the actual values required would make the representation of the steady shear and dynamic data less than adequate. If one ignores this objection, and proceeds to compare the relaxation data with the model predictions, the following issues emerge.

The non-linear spring coupled with (arbitrary) relaxation modes is certainly capable of mimicking the rapid stress drop from the steady state observed in Fig. 12a and b. When such a comparison is made at lower strains (i.e. at lower stress levels than at the steady state), the model does not predict a rapid decrease presumably because the chain extension is such that the spring is now in a linear region. However, the experimental data clearly indicate that even under these conditions, there does exist a raid decrease in the stress levels (see also Figs. 1 and 12a).


In summary, the experimental data may be explained within the framework of the FENE-P model, but only with arbitrary or at least less than satisfactory means of choosing parameter values. A more pragmatic possibility is the inclusion of a specific viscous dissipation term in the dumbbell models. Initial suggestions in this direction have been made by Hinch [15] and more recently by Rallison [16]. More experiments are needed, however, to completely resolve this matter.

a. .,,.., w w .=

U)

1 E+06

1 E+05

1 E+04

1 E+03

1E+06

IE+05

1E+04

1E+03 2.25

1 2.45 2.65 2.85

o Total Stress Data

• Elastic Stress

FENE-P

. . . . . . EquaUon (11)

1E+02

1E+01 I 0 0.5 1 1.5 2 2.5 3

t i m e ( s e c )

Fig. ll(a). Total and elastic data compared with FENE-P predictions for fluid A at a strain rate of 2.1 s -]. The dotted line shows the predictions using Eq. (11) with the FENE-P model. Inset shows detailed view of stress growth near steady state.


• - . 1E+04 a a .

E

1E+03

1E+02

1E+01

1 E+06

II 1E+05

II 1E+03 r r 1., 1., 1., 1., i i F

0

o

0 0.5

o To ta l S t r e s s Data

• E las t i c Stresss

1.5 2

• •

1.g 2 ~ / •

I _d, ,8 '7

time (sec)

Fig. 1 l(b). Total and elastic data compared with FENE-P predictions for fluid A at a strain rate of 3 s-i. Inset shows detailed view of stress growth near steady state.

4. Conclusions

It has been demonstrated that, at the cessation of extensional flow, the filament diameter remains constant for several seconds, enabling relaxation measurements to be conducted on the fluid sample.

These experiments show that large viscous stresses arise during stretching of dilute polymer solutions. The viscous stresses were observed to be dependent on the actual value of elastic stress (i.e. chain extension) and strain rate. The elastic stress at steady state was observed to be independent o f strain rate.

Using parameters established from shear data, the F E N E - P model was shown to por t ray the growth of the elastic stress. Inclusion of a viscous dissipative stress into this model allowed the


A m n

4.1 u)

1 E+06

1 E+05

1E+04

1 E+03

1E+02

1E+01

1E+04

1 E +03

1.11

D

_I

I 75 2 225 2 ~

.,,.% FENE-P Predictions

0 1 2 3 4

t ime (sec)

Fig. 12(a). Stress growth and subsequent relaxation compared with FENE-P predictions for fluid A at a strain rate of 2.1 s - m. Inset shows details of stress growth and decay for the smallest value of chain extension. Note that stress jump is present for values of chain extension other than steady state.

total stress to be predicted. Further work is needed to better understand the physics which leads to this dissipative stress.

The steady state values of stress obtained after transient stress growth were observed to be path independent, and a new technique has been introduced which gives excellent agreement between radial and axial based strain rates. This result indicates that the majority of the fluid sample deforms homogeneously.

102 N.Y. Orr, T. Sridhar / J. Non-Newtonian Fluid Mech. 67 (1996) 77-103

1 E+06

A e: n

¢o w .o

4.m

¢n

1E+05

1E+04

1E+03

1E+02

1E+01

FENE-P Predict ions

0 1 2 3 4

t ime (sec)

Fig. 12(b). Stress growth and subsequent relaxation compared with FENE-P predictions for fluid A at a strain rate of 3 s - l .

Acknowledgements

This work was supported by a program grant from the Australian Research Council. The authors acknowledge discussions with Dr E.J. Hinch and Dr J. Rallison during the Newton Institute workshop on "Unresolved Experimental Dilemmas in the Dynamics of Complex Fluids". Discussions with Professor G. McKinley clarified many aspects of the FENE modelling of these data.


References

[1] R.K. Gupta and T. Sridhar. Elongational Rheometers, A.A. Collyer and D.W. Clegg (Eds.), Rheological Measurements, Elsevier, London (1988)

[2] D.F. James and K. Waiters, A critical appraisal of available methods for the measurement of extensional properties of mobile systems. A.A. Collyer (Ed), Techniques in Rheological Measurements, Elsevier, London (1993)

[3] V. Tirtaatmadja and T. Sridhar, Comparison of constitutive equations for polymer solutions in uniaxial extension, J. Rheol., 39 (1995) 1133-1160

[4] R.G. Larson, Constitutive equations for polymer melts and solutions, Butterworths, Boston, 1988. [5] C.W. Manke and M.C. Williams, Stress jump at the inception of shear and elongational flows of dilute polymer

solutions due to internal viscosity, J. Rheol., 31 (1987) 495-510. [6] C.W. Manke and M.C. Williams, Comparison of a new internal viscosity model with other constrained-connec-

tor theories of dilute polymer solution rheology, Rheol. Acts, 32 (1993) 418-421. [7] C.H. Liang, and M.E. Mackay, The stress jump of a semi-rigid macromolecule after shear: steady state results,

J. Rheol., 37 (1993) 149-174. [8] S.F. Smyth, C.H. Liang, M.E. Mackay, and G.G. Fuller, The stress jump of a semi-rigid macromolecule after

shear: comparison of the elastic stress to the birefringence, J. Rheol., 39 (1995) 659-672. [9] C.C. Hua and J.D. Schieber, Non-equilibrium brownian dynamics of Hookean and FENE dumbbells with

internal viscosity, J. Non-Newtonian Fluid Mech., 56 (1995) 307-332. [10] D.F. James, and J.H. Saringer, Extensional flow of dilute polymer solutions, J. Fluid Mech., 97 (1980) 655-671. [11] D.H. King, and D.F., James, Analysis of the Rouse model in extensional flow II, J. Chem. Phys., 78 (1983)

4749-4754. [12] G. Ryskin, Calculation of the effect of polymer additive in a converging flow, J. Fluid Mech., 178 (1987)

423 -440 [13] J.M. Rallison, and E.J. Hinch, Do we understand the physics in the constitutive equation?, J. Non-Newtonian

Fluid Mech., 29 (1988) 37-55. [14] R.G. Larson, The unravelling of a polymer chain in a strong extensional flow, Rheol. Acta, 29 (1990) 371-384. [15] E.J. Hinch, Uncoiling a polymer molecule in a strong extensional flow, J. Non-Newtonian Fluid Mech., 54

(1994) 209-230. [16] J.M. Rallison, Dissipative stresses in dilute polymer solutions, J. Non-Newtonian Fluid Mech., submitted for

publication, 1996. [17] D.F. James, and T. Sridhar, Molecular conformation during steady state measurements of extensional viscosity,

J. Rheol 39 (1995) 713-725. [18] G.K. Batchelor, The stress generated in a non-dilute suspension of elongated particles by pure straining motion,

J. Fluid Mech., 46(4) (1971) 813-829. [19] F. Brochard, and P.G. DeGennes, Dynamical scaling for polymers in theta solvents, Macromolecules, 10 (1977)

1157-1161. [20] A.V. Vologodskii, A.V. Lakashin, M.D. Frank-Kamentskii and V.V. Ansaelevich, The knot problem in

statistical mechanics of polymer chain, Sov. Phys, JETP (Engl. Transl.), 39 (1975) 1059-1063. [21] V. Tirtaatmadja and T. Sridhar, A filament stretching device for measurement of extensional viscosity, J. Rheol.,

37 (1993) 1081-1102. [22] V. Tirtaatmadja, Measurement of extensional viscosity of polymer solutions, PhD Thesis, Monash University,

Clayton, Vic. [23] S.H. Spiegelberg and G.H. McKinley, J. Non-Newtonian Fluid Mech., 67 (1996) 49-76.

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Stress relaxation in uniaxial extension