Rheology for Polymer Melt Processing

RHEOLOGY FOR POLYMER MELT PROCESSING RHEOLOGY SERIES Advisory Editor: K. Waiters FRS, Professor of Applied Mathematic...

Author: Piau J.-M. | Agassant J.-F.

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RHEOLOGY FOR POLYMER MELT PROCESSING

RHEOLOGY SERIES Advisory Editor: K. Waiters FRS, Professor of Applied Mathematics, University of Wales, Aberystwyth, U.K.

Vol.

1 Numerical Simulation of Non-Newtonian Flow (M.J. Crochet, A.R. Davies and K. Waiters)

Vol.

2 Rheology of Materials and Engineering Structures (Z. Sobotka)

Vol.

3 An Introduction to Rheology (H.A. Barnes, J.F. Hutton and K. Waiters)

Vol.

4 Rheological Phenomena in Focus (D.V. Boger and K. Waiters)

Vol.

5 Rheology for Polymer Melt Processing (Edited by J-M. Piau and J-F. Agassant)

RHEOLOGY FOR POLYMER MELT PROCESSING

Edited by

J-M. Piau

Laboratoire de Rheologie, Domaine Universitaire, Grenoble, France

and

J-F. Agassant

CEMEF Ecole des Mines, Valbonne Cedex, France

1996 Elsevier Amsterdam

- Lausanne

- New

York - Oxford

- Shannon

- Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

Library

oF Congress C a t a l o g i n g - i n - P u b l i c a t i o n

Data

Rheology f o r polymer melt process|ng / [ e d i t e d by] J.-M. Piau and d. -F. Agassant. p. cm. - - (Rheo]ogy s e r i e s ; v o l . 5) Inc]udes b i b l i o g r a p h i c a l r e f e r e n c e s and index. ISBN 0-444-82236-4 ( a l k . paper) 1. Polymers--Rheology. 2. Po]ymer m e l t i n g . I . Piau. J.-M. I I . Agassant, J . - F . I I I . S e r l e s : Rheology s e r i e s ; 5. TP1150.R49 1996 668.9--dc20 96-31377 CIP

ISBN: 0 444 82236 4 91996 ELSEVIER SCIENCE B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U . S . A . - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands.

PREFACE

This book presents the main results obtained by different laboratories involved in the research group "Rheology for polymer melts processing" and belonging to various French universities, schools of engineering and to the CNRS (Centre National de la Recherche Scientifique - France). This research group was created in 1987 and is supported by the CNRS, the French Ministry for Research and Technology and three major French companies (ELF-ATOCHEM, MICHELIN and RHONE-POULENC). The group has comprised up to 15 research laboratories with different skills (chemistry, physics, material sciences, mechanics, mathematics) but with a common challenge : to make progress in understanding the relationships between macromolecule species, their rheology and their processing. The first problem was to find a language common to all the experts in these different fields in order to promote effective cooperations. This was achieved through regular technical meetings (a minimum of two meetings every year) as well as open meetings (in Paris, Sophia Antipolis, ViUard de Lans, Biarritz, Le Mans, Strasbourg, and Paris again for the final meeting) where recognized international scientists were invited. Some crucial issues of polymer science have been addressed ; correlation of viscoelastic macroscopic bulk property measurements and models, slip at the wall, extrusion defects, correlation between numerical flow simulations and experiments. Significant research results have been obtained, and have been or will be published in International Journals or congress proceedings. In addition, every participant has benefited in his own research activities from being a member of the "Rheology for polymer processing" research group and is now better prepared for developing further cooperation as well as for addressing new problems. The selection of results provided in the present book is unique in the sense that it allows one to grasp the key issues in polymer rheology and processing at once, through a series of detailed state of the art contributions. Generally these issues can only be found in different books. Each paper was reviewed by experts and by the book editors and some coordination was established in order to achieve a readable and easy access style.

vi Nevertheless, each author remains responsible for his own contribution. Papers have been gathered in sections covering successively : "Molecular dynamics", "Constitutive equations and numerical modelling", "Simple and complex flows". However, each paper can be read independently. This book can be considered as an introduction to the main topics in polymer processing. It is therefore intended to be useful for graduate students as well as for scientists in academic or industrial research laboratories 9polymer suppliers now have the opportunity through the development of new catalysts and new polymer blends to propose speciality polymers which are well adapted to a specific type of production but this entails better control of the relationships between macromolecular structure, rheology and processing. On the other hand, machine or tool makers want to define the best tools and processing conditions for obtaining maximum throughput free of defects, at a controlled final temperature, with minimum trial and error. Computer-assisted design is starting to be commonly used but this calls for strong numerical algorithms as well as realistic constitutive equations. Finally, polymer converters are often small companies but with a high level of innovation. They also need appropriate numerical software in order to choose the right polymer and machine to achieve the best product properties. We hope that this book will help to enhance the interest of the scientific and technical community for the fascinating field of rheology and polymer sciences. We are very proud to be surrounded by he enthusiastic colleagues of the "Rheology for polymer processing" research group and we thank them for their assistance in the preparation of this book.

J. M. Piau, J. F. Agassant

~176

VII

CONTENTS

PREFACE

.............................................................................

v

I. M O L E C U L A R DYNAMICS I. 1 The reptation model : tests through diffusion measurements in linear polymer melts L. l.,6ger, H. Hervet, P. Auroy, E. Boucher, G. Massey I n t r o d u c t i o n ..................................................................... T h e r e p t a t i o n m o d e l ............................................................ Diffusion m e a s u r e m e n t s in p o l y m e r systems ............................... Interpretation and comparison with rheometrical data ..................... Conclusions ....................................................................

1 2 6 11 15

1.2 Polybutadiene : NMR and Temporary elasticity J.P. Cohen Addad Introduction ..................................................................... T e m p o r a r y n e t w o r k structures ............................................... S e g m e n t a l motions : dynamic screening effect ............................. M o l t e n high p o l y m e r s 9 semi-local dynamics ............................... Conclusion ......................................................................

17 20 28 33 35

1.3 Chain relaxation processes of uniaxially stretched polymer chains : an infrared dichroism study J.F. Tassin, L. Bokobza, C. Hayes, L. Monnerie Introduction ..................................................................... Theoretical background in Infrared dichroism .............................. Experimental .................................................................... T h e o r e t i c a l basis o f interpretation ............................................ Results and discussion on isotopically labeled chains ..................... Results and discussion on isotopically labelled 6-arm stars .............. Results and discussion on binary blends of long and short chains ....... Conclusion ......................................................................

37 38 39 41 44 49 55 61

1.4 Chain conformation in elongational and shear flow as seen by SANS R. Muller, C. Picot I n t r o d u c t i o n ..................................................................... Methodology .................................................................... E l o n g a t i o n a l f l o w .............................................................. S h e a r f l o w ...................................................................... C o n c l u s i o n s and p e r s p e c t i v e s ................................................

65 66 73 87 93

~176 VIII

1.5 Molecular rheology and linear viscoelasticity G. Marrin, J.P. Montfort Introduction ..................................................................... Linear viscoelastic behaviour of linear and flexible chains - basics and p h e n o m e n o l o g y ............................................................. The case of entangled monodisperse linear species : pure reptafion ...... E n t a n g l e d m o d e l - b r a n c h e d p o l y m e r s ........................................ Entangled polydisperse linear chains : double reptation ................... Effects of non e n t a n g l e d chains .............................................. P r o b l e m s still p e n d i n g .........................................................

95 96 105 114 119 129 135

II. CONSTITUTIVE EQUATIONS AND N U M E R I C A L M O D E L L I N G II. 1 Experimental validation of non linear network models C. Carrot, J. Guillet, P. Revenu, A. Arsac Introduction ..................................................................... T h e o r e t i c a l aspects ............................................................. E x p e r i m e n t a l aspects ........................................................... Experimental validation of the W a g n e r m o d e l ............................... Experimental validation of the Phan Thien-Tanner model ................... Conclusion ......................................................................

141 144 159 167 176 190

II.2 Mathematical analysis of differential models for viscoelastic fluids J. Baranger, C. Guillop6, J.C. Saut I n t r o d u c t i o n . T h e m o d e l s ..................................................... Maxwell type models : loss of evolution and change of type ............. S t e a d y f l o w s .................................................................... U n s t e a d y f l o w s ................................................................. S t a b i l i t y i s s u e s .................................................................. Numerical analysis of viscoelastic flows .................................... Conclusion ......................................................................

199 201 203 208 214 225 230

II.3 Computation of 2D viscoelastic flows for a differential constitutive equation Y. Demay Introduction ..................................................................... Computation of a purely viscous flow ....................................... Finite elements method for viscoelastic flows .............................. Application ...................................................................... Conclusion ......................................................................

237 240 244 252 252

III. SIMPLE AND COMPLEX FLOWS III. 1 Validity of the stress optical law and application of birefringence to polymer complex flows R. Muller, B. Vergnes Introduction ..................................................................... General relationships and usefulness of birefringence measurements ... Validity of the stress optical law .............................................. Application to complex flow studies ......................................... Conclusion ......................................................................

257 257 264 277 281

III.2 Comparison between experimental data and numerical models J. Guillet, C. Carrot, B.S. Kim, J.F. Agassant, B. Vergnes, C. B6raudo, J.R. Clermont, M. Normandin, Y. B6raux Introduction ..................................................................... Materials ......................................................................... C o n s t i t u t i v e e q u a t i o n s ......................................................... F l o w g e o m e t r i e s and e x p e r i m e n t s ............................................ Numerical models .............................................................. C o m p a r i s o n between numerical results and experiments .................. Conclusions .....................................................................

285 289 289 295 300 317 333

111.3 Slip at the wall L. IAger, H. Hervet, G. Massey Introduction ..................................................................... Local determination of the velocity at the wall .............................. M o l e c u l a r m o d e l s and discussion ............................................ Conclusions ....................................................................

337 338 348 353

III.4 Slip and friction of polymer melt flows N. E1 Kissi, J.M. Piau Introduction ..................................................................... Means used ..................................................................... F l o w in h i g h surface e n e r g y dies ............................................ F l o w in dies with low surface energy ....................................... D i s c u s s i o n and c o n c l u s i o n ....................................................

357 359 361 372 384

III.5 Stability phenomena during polymer melt extrusion N. E1 Kissi, J.M. Piau Introduction ..................................................................... E x p e r i m e n t a l facilities and flow curves ..................................... V i s u a l i z a t i o n of u p s t r e a m flow ............................................... O b s e r v a t i o n of stable flow - sharkskin ...................................... Observation of unstable flow for slightly to moderately entangled p o l y m e r s - melt fracture .................................................... H i g h l y e n t a n g l e d polymers - flow with slip ................................ Conclusion ...................................................................... SUBJECT

INDEX

....................................................................

389 391 397 402 408 413 415 421

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Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) 1996 Elsevier Science B.V.

T h e reptation model: tests t h r o u g h d i f f u s i o n m e a s u r e m e n t s in l i n e a r p o l y m e r melts

L. LEGER, H. HERVET, P. AUROY, E. BOUCHER, G. MASSEY, Laboratoire de Physique de la Mati~re Condens6e, URA CNRS 792, Coll~ge de France, 11 Place Marcelin-Berthelot, 75231 PARIS Cedex 05, FRANCE 1. INTRODUCTION The wide range of applications of polymer materials relies in part on their unique viscoelastic behaviour. In the melt state, a polymer sample can be strongly deformed (deformation ratio of several hundred of percent) without breakage, and then recover its original shape almost completely if the stress is relaxed after a short enough time. This memory is completely lost after long periods of time, and the same material flows like any ordinary viscous liquid. A number of methods have been developed to investigate this behaviour [ 1] and to try to relate the time after which the system behaves like a simple liquid to its molecular characteristics. The Brownian motion, in polymer melts or in sufficiently concentrated solutions of linear flexible macromolecules, also presents unique features, the most well known being a self diffusion coefficient scaling as Mw -2, with Mw the weight average molecular weight of the polymer chains [2], for large molecular weights. Viscoelasticity (response to an external mechanical excitation) and diffusion (response to thermal fluctuations) must have their origin in the same dynamical process at the molecular level, and any model proposed to explain one aspect must also consistently account for the other. Early in the development of this area it was suggested that the unique dynamics of dense long polymer chains was due to molecular "entanglements" [3], first introduced as a rather loose concept which expresses the strong restrictions that uncrossable, unidimensional objects must exert on the motion of their neighbours. The reptation model, proposed by P.G. de Gennes in 1971 [4], gives a framework to describe the motion of entangled chains, and leads to a universal description of the linear viscoelasticity (description developed by S. Edwards and M. Doi [5, 6]) of long flexible polymer systems. One interesting feature of this model is that it provides laws for the variation of the key physical quantities of the linear viscoelasticity as for example the zero shear viscosity, 1"10,or the terminal relaxation time, Tt, with the molecular parameters of the system such as the polymerisation index of the chains, N, (related to the molecular weight through M = Nm, with m the monomer molecular weight) and the polymer volume fraction if one is dealing with a solution. These laws are in very good qualitative agreement with the large number of available experimental data on the dynamic behaviour of linear flexible entangled polymers, but quantitative departures still remain between the experiments and the predictions, as for example the fact that the exponent of the power law which characterises the variation of the zero shear viscosity with the molecular weight is observed to be 3.3 or 3.4 [3, 6] rather than the predicted value 3. These deviations have lead to a long controversy on the validity of the reptation model, and have stimulated a series of experimental and theoretical investigations to try to understand the limitations of this model and to propose the necessary modifications to obtain a better description of the dynamic properties of liquid polymers [7 to 22].

In the present paper, after a rapid presentation of the reptation model in its simplest version, in order to pinpoint the underlying hypothesis, we discuss the interest of complementary self diffusion and viscoelastic measurements, and present the currently available methods for measuring diffusion in entangled polymer systems. Then, results obtained on polydimethylsiloxane (PDMS), a model liquid polymer well above its glass temperaturd at room temperature will be described, and the consequences on the limits of the entangled regime as seen from diffusion measurements, compared to what is observed in rheometry, will be discussed. 2. THE REPTATION MODEL: 2.1. One linear chain among fixed obstacles: The Brownian diffusion of one long polymer chain trapped among fixed obstacles has been described quantitatively by P.G. de Gennes in 1971 [4]. As schematically presented in Fig. l a, the chain cannot cross the obstacles. At any time, its shape depends on the actual monomers-obstacles interactions: the chain is confined by the obstacles. The only way for the chain to change its configuration is to find a path among the obstacles through small fluctuations of its contour, folding or unfolding locally. The leading motion takes place at the 9 9

9

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9

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.

-

.

_b

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Figure 1" Schematic representation of one chain among obstacles, a) The chain is constrained by the obstacles, b) By local fluctuations, the chain changes its conformation. The probability of forming a loop (dashed line) is very small (strong entropy loss), and the role of the extemities is dominant, c) The chain reptates like a snake in the virtual tube (thin line) envelope of all the topological constraints exerted on it by the obstacles. The tube is progressively redefined from the extremities, as schematically presented by the two situations at time t and t', with t'>t.

extremities of the chain, as a consequence of the low probability for the chain of forming a long loop at any other central part (figure tb), the formation of a large loop implying a large entropy loss (the two sides of the loop have to follow the same path among the obstacles). time. As a result, the extremities can engage between new obstacles and progressively pull the chain in a new environment (figure l c). This is the reptation process. As a consequence, the central part of the chain remains trapped by the same topological constraints over a long period of A way of taking these constraints into account in an averaged manner was proposed by S. Edwards [23]: one can assume that the chain is trapped in a virtual tube, or envelope made of all the obstacles which directly surround it (thin line on figure l c). The chain can move freely along the curvilinear axis of the tube, without encountering obstacles, while it cannot escape out of the tube laterally. At any time, by fluctuations of its local kinks, the chain leaves some parts of the tube, and creates new parts. The detailed statistical description of the process [4] leads to definite predictions for the molecular weight dependence of the diffusion coefficient, that can be reconstructed using simple arguments that we present now. The chain is assumed to be an ideal Gaussian chain made of N monomers of size a (that is indeed the case in a polymer melt [24]) and its average end to end vector has a length R0 = Nl/2a. For a free ideal chain, the dynamics has been modelled by Rouse [25], assuming that the chain is similar to a collection of springs, connected by beads with the size of a monomer, where all the friction is concentrated. This leads to a longest relaxation time of the chain proportional to N 2, and to a translational diffusion coefficient of the centre of mass inversely proportional to N. In the presence of the obstacles, the local fluctuations of the chain contour which involve distances smaller than the diameter of the tube, d, are not affected by the tube. The chain can thus be considered as a necklace of beads of size d, the average distance between the obstacles. The corresponding average number of monomers inside one bead is Nd, with d = Ndl/2a. The portion of chain inside a bead obeys Rouse-like dynamics, because the monomers inside the bead are not affected by the tube. The longest relaxation time of one bead is "eR(d) = ZlNd 2 ,

(1)

and the corresponding diffusion coefficient is D d -- ~,D1 Nd

(2)

where ~1 and D1 are the monomer characteristic time and diffusion coefficient respectively. "el, D! and "eR, DR are related by general diffusion laws" "elDl -- a 2, "eR(d)Dd = d 2.

(3)

The mobility of the whole chain, free to move along the curvilinear axis of the tube is N/Nd smaller than the mobility of one bead, as the friction on the full necklace is the friction on one bead times the number of beads. The diffusion coefficient of the chain along the tube is D t ~ ~DI N

(4)

The reptation time, TR, or the time it takes for the chain to renew its configuration is related to Dt through a relation analogous to eq.3: DtTR = Lt 2, with Lt = d(N/Nd) the total length of the tube. Thus

TR

"t:d(d)

(%)3

d

= %1

NfifN

d

9

(5)

TR is the longest relaxation time of the chain constrained by the obstacles. It is much longer than the longest Rouse time of the free chain, z R (N) --- xlN 2. It should be noticed that the measurable self diffusion coefficient Ds of the chain is not Dr: the tube is contorted, with a Gaussian configuration, and when the chain travels a distance Lt along the tube it only travels a distance R0 in a given direction of the real space, so that DsT R -- Ro 2 = Na 2, and Ds

N d -- D 1

N 2" (6)

The diffusion is much slower (by a factor N) than for a Rouse-type free chain. 2.2. Polymer melts: It is tempting to apply the ideas developed in section 2.1. to describe the dynamics of long linear polymer molecules in the melt state. The average radius of the chain is R0 -- N 1/2a [24]. The volume spanned by one chain, R03, is much larger, for long chains, than the volume effectively filled by the monomers of that chain, l) -- Na 3. In the melt state where the monomers are closely packed, the chains are thus interpenetrated. Since they cannot cross each other, they are strongly constrained, in a way somewhat similar to the situation of one chain among the fixed obstacles of section 2.1. One can again define a tube, envelope of all the topological constraints exerted on one chain by its surrounding neighbours. But one is now faced with two major problems. 1) All the chains in the system move, and the obstacles are not permanent. In order to apply reptation ideas to a polymer melt, it is necessary to assume or to establish that the evolution of the tube due to the motions of all the surrounding chains, is slower than the reptation. 2) The tube diameter is no longer an external parameter, it represents the average distance perpendicular to the local chain direction that one monomer can travel, due to the local chain flexibility, before being blocked by the surrounding chains. A complete determination would require a description of the actual monomer-monomer interactions and is still out of reach, despite strong efforts to do so [19, 26 to 28], and at present it is not fully understood what factors determine the tube diameter in polymer melts. It has been introduced as a phenomenological parameter, through an average number of monomers necessary to get an entanglement, Ne, with d -- Nel/2a. This is in fact the definition of an entanglement: it is not a knot between two chains, but an ensemble of constraints which the chains collectivelly exert on each other, so that the diffusive motions of the monomers are no longer isotropic, when the explored distance is large enough. Then, both the longest relaxation time TR and the self diffusion coefficient can be estimated through eq. 5 and 6 respectively, replacing Nd by Ne. One gets: TR ='1:1 N~N e'

(7)

D s = D ! Ne///2.

(8)

If chains with a polymerisation index smaller than Ne are used, they are no longer efficiently

constrained by their neighbours, and the reptation picture no longer holds. A Rouse-like dynamics should be recovered, with a self diffusion coefficient proportional to 1/N. When reptation is used to develop a description of the linear viscoelasticity of polymer melts [5, 6], the same underlying hypothesis ismade, and the same phenomenological parameter Ne appears. Basically, to describe the relaxation after a step strain, for example, each chain is assumed to first reorganise inside its deformed tube, with a Rouse-like dynamics, and then to slowly return to isotropy, relaxing the deformed tube by reptation (see the paper by Montfort et al in this book). Along these lines, the plateau relaxation modulus, the steady state compliance and the zero shear viscosity should be respectively: GN0=

J0=

kT Nea3 ' 6

5GN ~ ~2 no = ~ G N O T R 9

(9)

(10) (11)

Of course, all these relations are expected to be valid only if the reptation description holds, i.e. if the motion of the tube due to the dynamics of all the surrounding chains is much slower than the reptation of one chain. Eq. I 1 provides an easy way of checking the validity of the reptation model: the zero shear viscosity should depend on the polymerisation index of the chains like N 3. Experimentally, the observed exponent is larger, 3.3 to 3.5, except perhaps when extremely large molecular weights are used [29]. The reason for the discrepancy has been debated for many years and remains not totally elucidated. It is indeed a puzzling question: the dependences of the storage modulus versus frequency, for various molecular weights have been observed to agree well with the Doi-Edwards predictions [30], and the molecular weight dependence of the self diffusion coefficient has often been observed to agree well with the reptation prediction [2, 31 to 35]. Moreover,. more local tests of the reptation dynamics, as for example determinations of the monomer concentration profiles in a macroscopic diffusion experiment starting with a step-like labelled chains profile, through neutrons techniques, appear to agree very well with the reptation picture [34, 36, 37]. The question of the detailed limits of validity of the reptation model thus remains a pending question. What appears puzzling is the fact that, on one hand, the reptation model and the Doi - Edwards' description of the linear viscoelasticity work so well both qualitatively and quantitatively for some experiments, while, on the other hand, they seem unable to account for all the existing data. This may suggest that the reptation model does not contain the whole story of linear polymer dynamics, and that one needs to learn more on other possibly competing processes. The obviously weak points in the use of reptation ideas to describe the dynamics of linear polymer melts are the two hypotheses mentioned above. Their validity is not easy to check experimentally and it is also not easy to understand how to relate Ne to the molecular properties of the polymer. In fact, it is not simple to determine Ne in a reliable manner: prefactors not taken into account in the simple arguments leading to eq. 7 to 11 may exist and the question is not a trivial one. Severalroutes are possible: one relies on measurements of GN 0. A strong effort has been made recently by L.J. Fetters and coworkers [38] to relate the value of the critical molecular weight between entanglements, as deduced from the plateau modulus, to the molecular structure of the polymer. The correlations they obtain are remarkable and should allow one to predict how one can imagine to define a given chemistry in order to reach a given rheology. Such correlations rely on measurements performed on high molecular weight polymers, well above Ne, i.e. highly entangled. Another possible route

consists in following the well-admitted idea that a way of estimating Ne is to locate the crossover between entangled and non-entangled regimes by looking for the appearance of a plateau modulus when the molecular weight is progressively increased, or for a change of slope in a log-log plot of the zero shear viscosity versus the molecular weight. It is not clear however that this is the best way to do so: decreasing the molecular weight accelerates the dynamics of all the chains in the media, and it may well be that the crossover region could not be described, even qualitatively by reptation ideas. When decreasing the molecular weight one can eventually enter into a regime in which the chains are still entangled i.e. dynamically constrainted by each other, but in which the reptation hypothesis is no longer valid, due to the onset of collective motions of the chains. The reptation picture is a one chain picture, the topological constraints exerted on one chain by its neighbours being taken into account in an average way, through the tube notion. Collective effects are not neglected in this framework, and if they become the dominant dynamical process, the reptation model no longer holds. To try to characterise experimentally the importance of the collective effects on the dynamics of one chain, diffusion measurements reveal to be a unique tool. Mixtures of long and short chains can be used, along with labelling techniques which permits one to follow the diffusion of either the short or the long chains. Thus, diffusion measurements allow one to decouple the question of the dynamics of the surrounding chains and the question of the cross over between entangled and non-entangled behaviour. When few short chains (index of polymerisation N) are immersed into much larger chains (index of polymerisation P), one can expect that for large enough P, the motions of the surrounding chains will be frozen down on the time scale of the motion of the test chains. These labelling techniques should enable one to characterise the crossover region between entangled and disentangled behaviour, by measuring the diffusion coefficient as a function of N, in situations where the motions of the environment remain slow (P>>N). Then, the importance on the dynamics of the collective motions can be characterised, varying the ratio P/N, and comparing, at fixed N, the data for N = P and N > Ne, in good agreement with experiments. However, for N close to Ne the above description certainly no longer holds: all the chains in the system move by both reptation and constraint release, and their motions are accelerated compared to pure reptation, due to the additionnal degrees of freedom of tube renewal. The constraint release process can no longer be treated as a small perturbation to reptation. Attempts have been made to describe the motion in a self consistent way [ 10], but are not really satisfactory: they lead to an unphysical divergence of the characteristic time of the motions of the chains and are unable to correctly describe the crossover towards the nonentangled dynamical regime, because they do not introduce the Rouse dynamics as another competting process. This question has been addressed recently by D. Pearson et al [49], with an extensive investigation of both the self-diffusion and the zero shear viscosity as a function of molecular weight in polyethylene samples of low polydispersity. The technique used to measure the self-diffusion coefficient, pulse field gradient NMR does not allow for measurements at fixed matrix, and the data of reference 49 have to be compared with our data for P = N. Trends very similar to what we have observed in PDMS are clearly seen in polyethylene, i.e. a regime at large molecular weights well described by simple reptation arguments (Ds --- N -2) and an acceleration of the diffusion at lower molecular weights, associated with additional degrees of freedom such as tube renewal or possibly fluctuations of the test chains inside their tube. The important point made by Pearson et al is that the crossover between the Rouse-like regime and the entangled one is most clearly evidenced if one reports the product riDs, which is independent of the local friction, as a function of the molecular weight: in the Rouse dynamical regime this product is expected to be independent of molecular weight, while it should increase linearly with Mw, in the pure reptation regime. In order to perform the same kind of analysis of our data, we have measured the zero shear viscosity of our PDMS sample. These measurements have been kindly performed for us by R. Muller from I.C.S. Strasbourg, and are reported, along with low molecular weights data from reference 45 in Fig. 5. In Fig. 6, we have reported the product riD, for PDMS, as a function of the polymer molecular weight. One has to notice that for the viscosity measurements one is always in a

13

situation with N = P, and thus the filled symbols cannot be interpreted as corresponding to chains moving in a frozen environment. In a way very similar to what has been obtained in polyethylene, we observe a low molecular weight regime which could be Rouse-like, and a transition towards an entangled regime for both sets of data. 10 4

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I

I IIII

10 4

N

Figure 5: Zero shear viscosity as a function of the polymerisation index of the chains for PDMS at T = 27~ The data up to N = 400 are from reference 45, while the larger molecular weights have been measured at I.C.S. Strasbourg by R. Muller and give an exponent 3.37 for the viscosity - molecular weight law. One can try to locate a critical polymerisation index above which the data are no longer compatible with a Rouse-like dynamics, Ne' = 500, lager than the Ne = 100 value determined from the diffusion measurements in a frozen matrix. This is an illustration of the fact that the two processes, Rouse motion and entangled motion are in competition: the slowest process is the one which is indeed observed.When the matrix chains are mobile, the entangled dynamics becomes more rapid than pure reptation, and the Rouse motion can dominate the dynamics for larger molecular weights than when the matrix chains are immobile. In fact, in the crossover region, the chains are still entangled (this appears on the diffusion data with N

=

0,6

~q

-~

0,4

0

-- . . . .

-5

:

-4

. . . .

I

-3

. . . .

I

-2

. . . .

I

-1

,

_

0

1

2

log t / xe Figure 7. Relative relaxation of chain-end of PS D H D 500 ( 0 ) w i t h e s t i m a t e d error bars compared to the prediction of fluctuation, retraction and reptation processes. T h e s e results, as compared to a previous study [29], illustrate a better a g r e e m e n t b e t w e e n theory and experiments. This can be attributed to a better a p p r o x i m a t i o n of the mode d i s t r i b u t i o n i n v o l v e d in the chain f l u c t u a t i o n process and to which the end of the chains are quite sensitive. T h e s e data also prove t h a t u n d e r our e x p e r i m e n t a l conditions, the relaxation of the chain is d o m i n a t e d by the chain retraction and c h a i n - l e n g t h fluctuation process. The

49 influence of reptation is very weak and only the first modes contribute to the relaxation of chain ends.

1,2 ~9 ~0 ;>

0,8

c,t

0,6 ~9

0,4

eq

0,2

-5

-4

-3

-2

-1

0

1

2

log t / gB Figure 8. Relative relaxation of chain end of PS HDH 188 (~) with estimated error bars compared to the prediction of fluctuation, retraction and reptation processes. These experimental results which validate the basic processes of the Doi and Edwards model are in agreement with findings of Ylitalo et al. [30] who have been able to draw the same conclusions in step-shear flow experiments. Relaxation of a centrally deuterated chain in a high molecular weight matrix has led to the same conclusions [31]. However, other a t t e m p t s have not been able to distinguish between the relaxation of different parts of chains [32,33]. Discrepancy between the results of the various experiments m a y be partly due to a different contrast between the relative length of the blocks. Clearly, most discriminating results are obtained w h e n the investigated blocks are r a t h e r small, as it is particularly apparent in ref 30. 6. R e s u l t s a n d d i s c u s s i o n on isotopically l a b e l l e d 6-arm stars

In this section, we will compare results obtained on isotopically labelled p o l y s t y r e n e stars with those of the l i n e a r copolymer. With r e g a r d to the characteristics of the star, we may regard the linear polystyrene of molecular weight 180 k as having roughly the length of one arm (PS DH 184), the linear polystyrene of Mw = 500 k as having roughly the lengths of two arms (PS DHD 500) and a pure homopolymer of molecular weight 1.19 106 g/mol as having roughly the same molecular weight as the overall star (PS 1190). Inside a star, we m a y differentiate between the segments at the branch point (deuterated monomers located close to the branch point in star b), those of the end of an arm (deuterated monomers located at the ends of the arms of star b), and those of the long central block of star-a (hydrogenated monomers of star-a).

50 The results are reported as a function of t/x A since xA is independent of the molecular weight a n d n a t u r e (linear or s t a r shaped) of the polymer. This avoids the a r b i t r a r y choice of a reference temperature of various part~ of a star The orientational relaxation of various parts of the s t a r is depicted in Figure 9. Segments close to the branch point appear initially more oriented t h a n those of the central block. The end part of an a r m is clearly less oriented at short times. As relaxation proceeds, no difference can be made between the r e l a x a t i o n of the b r a n c h point and of the central block. The end block nevertheless relaxes faster. 6.1. O r i e n t a t i o n

6.2. C o m p a r i s o n b e t w e e n ~ a n d linear c h a i n s The average orientation of the 6-arm star is compared to t h a t of linear chains PS DH 184, PS DHD 500 and PS 1190 in Figure 10. It seems quite clear t h a t the short-time behaviour is almost the same in linear and star materials, i n d e p e n d e n t l y of the molecular weight. In contrast, the relaxation of the star, at longer times, is intermediate between t h a t of the linear chain PS DHD 500 and t h a t of PS 1190. It is therefore more rapid than t h a t of a linear chain of the same overall molecular weight but slower t h a n that of an unbranched chain of the same length (PS DHD 500). This suggests that long relaxation times are increased or even suppressed by branching.

0,2 e, 9 "',0

0,15

9

,

A O

0,1

~9

(D.

r

V

0,05

~

""O... ,

,-1

-0,5

0

0,5

1

1,5

2

2,5

3

log t/XA Figure 9. Relaxation of various parts of a star" branch point (o), central block (o) and chain end (~) In order to test whether the chain ends or the central part are responsible for this behaviour, the orientation of both blocks has been compared to t h a t of l i n e a r chains. The r e l a x a t i o n of chains ends in the s t a r polymer and in copolymers PS DH 184 and PS DHD 500 is compared in Figure 11. The short time orientation is similar for the 3 materials, but the end of an arm relaxes slower t h a n t h a t of the PS DHD 500 copolymer although the involvec( length is

51 slightly lower and the length of the arm also lower t h a n the half of the linear chain.

0,2

. . . .

I

. . . .

I

. . . .

I

. . . .

1

. . . .

x

0,15

:,

\

A r 9 r

0,1

n ~

m--j

r

V

~ . _ X x

0,05

9.

9 X

"'n.

-. . . . . . . . . X X .

X

-

9 g

-1

0

1

2 log t/~g

3

4

Figure 10. Comparison between the relaxation of average orientation of a star (x), PS DH 184 (o), PS DHD 500 (O) and PS 1190 (m) As far as the orientation of the central block is concerned, a slower relaxation is also observed for the star as shown in Figure 12. At long times, the central block of the star remains more oriented t h a n t h a t of the linear chain. The differences are significant and cannot be explained by simple differences between length of chains and blocks. This effect must therefore be a t t r i b u t e d to the "pinning" effect of the centre of the chain and s u b s e q u e n t suppression of the reptation motion. 0,2

/k

0,15 ~ ~

(2:) 9 r

V

0,1

,.,

-,

0,05

-~ I -1

0

1

2

3

4

log t/xA Figure 11. Relaxation of chain ends ofPS DH 184 (~1), PS DHD 500 (O), Star (0)

52

0 , 2

,

,

,

,

i

,

,

,

,

i

,

,

,

,

,

,

,

,

,

I

,

f~,

,

,

_

_ 9

-

0,15

-

A o

,

O 0,1

V

o,o5

9

I

-

o

L -1

' 0

1

2

--,,. 3

4

log t/xA Figure 12. Comparison between the relaxation of the central block (o) of the PS DHD 500 and of the central block of star (~) 6.3. Discussion

Before going into a detailed comparison of experiments with theoretical predictions, the main points of this study may be emphazised. The relaxation of the star is intermediate between t h a t of the PS DHD 500 and PS 1190 polymers. In this range of molecular weights, the viscosity and the t e r m i n a l relaxation times are highest for the star. This indicates t h a t under our e x p e r i m e n t a l conditions, the r e l a x a t i o n is not d o m i n a t e d by the d i s e n g a g e m e n t of the branches. However, the effect of pinning the centre of the chain is clear from the slower relaxation of the s t a r (or central block) as c o m p a r e d to the PS DHD 500 copolymer. The last point deals with the surprisingly higher orientation of the ends of the arms. Since the length of this p a r t of the arm is on the order of one tenth of the length of an arm, we expect a quite rapid relaxation, at least at much shorter times t h a n the arm fluctuation time. The strong orientation observed might be qualitatively interpreted as an effect of an orientational coupling with surrounding segments. The difference between s t a r and linear species would be due to a higher average orientation in the star as seen in Figure 10. As far as the fluctuation dynamics of an end-attached chain is concerned, a characteristic length is defined as the size of the t h e r m a l fluctuation of the tube which is given by : Leq (7) Xeq = # 2 v Narm/N e w h e r e N a r m / N e is the n u m b e r of e n t a n g l e m e n t s per arm and Leq is the equilibrium contour length of an arm, simply given by Leq = d Narm/Ne, d being the average contour length between entanglements. The associated relaxation time, ~eq, is given by :

53 d2 (Narm/Ne)2 2vkT

T,eq =

(8)

being t h e a v e r a g e friction coefficient a c t i n g on the d i s t a n c e d (~ = N e ~0)- T h e r e l a x a t i o n t i m e a s s o c i a t e d w i t h the complete r e l a x a t i o n of a n a r m is r e l a t e d to l:eq by "

~eq(

ll~ }112 exp(V Narm/Ne) (9) V Narm/Ne All t h e s e r e l a x a t i o n t i m e s can be e x p r e s s e d as a f u n c t i o n of t h e local r e l a x a t i o n t i m e ~A. S e h e m a t i e a l l y , at a n y t i m e following t h e s t e p d e f o r m a t i o n process, t h e s t r e s s is a s s u m e d to be r e l e a s e d only on t h e l e n g t h of a n a r m t h a t h a s been visited by a fluctuation. A r a p i d r e l a x a t i o n of c h a i n s e n d s is e x p e c t e d on t h e t i m e scale of Zeq for d i s t a n c e s s m a l l e r t h a n Xeq a n d t h e d y n a m i c s proposed by Viovy [13] c a n be u s e d as for l i n e a r chains. A t l o n g e r t i m e s , t h e fraction {(t) of a n a r m w h i c h h a s been r e l a x e d by a f l u c t u a t i o n c a n be c o m p u t e d as the solution of t h e following e q u a t i o n " Tm ----

t=

!

l:eq (v Na~m/Ne)nI/2 1__~exp(VNarm~Z/Ne)

(10)

a n d the s t r e s s is p r o p o r t i o n a l to (1 - ~(t)) for t i m e s such as Xeq < t < Xm 9Since our e x p e r i m e n t a l conditions correspond to r e l a x a t i o n t i m e s on t h e order of Zeq, t h e a p p r o x i m a t e e x p r e s s i o n for t h e r e l a x a t i o n at t i m e s l o n g e r t h a n Xm h a s not b e e n considered.

1,2

,

,

l

I

m l m m dmNlm m l ' I m m

'

mm m

'

'

~

I

'

'

'

I

'

'

'

I

'

'

J

)

...........................................

'

'

'

9

0,8 co ~9 0

0,6

["

0,2

.,..~

0 ,4

i

i

I

-02

, J , I , ~ , I , , , I .... -4

-2

0

2

I , , 4

6

8

log t / Z B F i g u r e 13. Theoretical orientation of various p a r t s of a s t a r : ( - - - ) end of a r m , ( ) a v e r a g e , ( ....... ) b r a n c h p o i n t a s s u m i n g r e l a x a t i o n by a r m fluctuation alone (light lines) or combined with r e t r a c t i o n (heavy lines).

54 It is thus possible to compute the relaxation of different p a r t s of the stars using this fluctuation process alone or combined with the retraction process. As an illustration, the orientational relaxation of a short end-block (like in stara) is compared to the average orientation and to the orientation of the branch point (like in star-b) in Figure 13. Since relevant relaxation processes occur on the time scale of XB, theoretical curves are presented with t/x B as abscissa. Although a very large difference can be expected if only the fluctuation process is t a k e n into account, the retraction of the arms increases significantly the overall relaxation and tends to induce less-differentiated behaviours. As far as q u a n t i t a t i v e c o m p a r i s o n with t h e e x p e r i m e n t a l d a t a is concerned, we decided to compare directly the e x p e r i m e n t a l l y d e t e r m i n e d orientation with the theoretical prediction. This requires a rescaling of the theoretical orientation in order to match it with the experimental data at times of the order of XA. A satisfactory a g r e e m e n t for the average orientation is observed as shown in Figure 14.

1,2 - , , , , ,

. . . .

I

. . . .

I ' ' ' ' 1 ' ' ' ' 1 ' ' ' "

1 r

"~ ~9

0,8

>

0,6

a,

0,4

[]

0,2 -4

-3

-2

-1 0 log t / x B

1

Figure 14. Comparison between the experimental rescaled average orientation of a star (~) with theoretical predictions (full line). Using the same rescaling, the orientation of the short end-block and of the central block of the s t a r can be compared with theoretical predictions as shown in Figures 15 and 16. This quantitative comparison between the theoretical and experimental relaxation of star polymers shows t h a t the main observations can be accounted for by the combination of the arm fluctuation and a r m retraction processes. A reasonable order of m a g n i t u d e of the orientation of chain ends can be theoretically predicted. However, the relaxation of the branch point is significantly more rapid t h a n predicted theoretically. This discrepancy m i g h t be a t t r i b u t e d to constraint-release processes which tend to relax the b r a n c h point of the star. The influence of the latter processes is more easily observed on the branch point where the relaxation should appear at longer times.

55

I

. . . .

I

'

"''

'

I

. . . .

0,8

0,6 =

0,4

q:~

0,2

[]

L 0 .

. . . . .

-4

,

. . . .

,

-3

. . . .

-2

, , .

.

.

.

.

.

.

-1 0 log t / ~B

.

.

1

2

Figure 15. Comparison between theoretical (full line) and experimental (el) relaxation of chain ends of star-a.

1,2

~

....

-,-,,

,D

,,,

,,1

0,8

....

, ....

i ....

~Ez

0,6 =

~

0,4

0,2 0

. . . .

-z

!

-3

. . . .

I

-2

,

~

~,,

I

-1 log t /

,

,

,

,

[

0

~

,

,

,

I

1

. . . .

2

"tB

Figure 16. Comparison between theoretical (full line) and experimental (a) relaxation of the branch point of star-b.

7. R e s u l t s a n d d i s c u s s i o n o n b i n a r y b l e n d s o f l o n g a n d s h o r t c h a i n s B i n a r y blends composed of d e u t e r a t e d short chains and of e n t a n g l e d h y d r o g e n a t e d long polystyrene chains have been investigated. The relaxational b e h a v i o u r of the s h o r t chains h a s been a n a l y s e d as a f u n c t i o n of t h e i r molecular weight. Additionally, a t t e n t i o n has been focussed on the role of the short chain concentration on the orientational relaxation of the long chains. F i g u r e 17 shows m a s t e r curves of the orientational r e l a x a t i o n of s h o r t d e u t e r a t e d chains of various molecular weights present at 20 wt.-% in a m a t r i x of molecular weight 2x106 g/mol (PS 2000). The m a s t e r curves have been d r a w n at a reference t e m p e r a t u r e of Tg + 9 ~ using t i m e - t e m p e r a t u r e superposition.

56 This choice relies on t h e a s s u m p t i o n t h a t a c o n s t a n t T g + T c o r r e s p o n d s to a c o n s t a n t free volume s t a t e . S u c h a n a p p r o x i m a t i o n p r e s u m e s t h a t the t h e r m a l e x p a n s i o n of free volume, af, as well as the fractional free volume, fg, at Tg, a r e i n d e p e n d e n t of t h e b l e n d composition. In t h e case of b i n a r y b l e n d s w i t h s h o r t chains, xA m a y d e p e n d on t h e composition of the blend, so t h a t a n o r m a l i s a t i o n of e x p e r i m e n t a l t i m e s w i t h t h i s p a r a m e t e r would be m i s l e a d i n g .

0.20

'"'',

....

u....

1 ....

I ....

I''''J

....

? 0.15

/b

A

O

\

~

r

\

xo

\

0.10

C)

~

.

%

r

0 \

\ o\

9

V

0.05

-

. ,~ ~ \ .,o~ ,-I. 9

0.00

,.,t -1

.... 0

~5-~ ----- 9 o- " * - " - - :c~ . o. _ o - o r - ~,b ~ ,,o,,, .... , ..... , .......... 1 2 3 4 5 log t (Tg+9~

.

TM-

6

F i g u r e 17. O r i e n t a t i o n a l r e l a x a t i o n of the s h o r t d e u t e r a t e d chains PSD72 (o), P S D 2 7 (O), PSD18 (i), PSD10 (~) a n d PSD3 (#), b l e n d e d a t 20 wt.-% with PS2000. While the o r i e n t a t i o n of t h e s e d e u t e r a t e d c h a i n s shows a s t r o n g m o l e c u l a r w e i g h t d e p e n d e n c e e s p e c i a l l y a t s h o r t t i m e s , all c h a i n s e x h i b i t a r e l a t i v e l y c o n s t a n t r e s i d u a l o r i e n t a t i o n a t long times. E v e n c h a i n s of m o l e c u l a r w e i g h t as low as 3000 p r e s e n t a n o n - z e r o local o r i e n t a t i o n . T h i s w a s c o n f i r m e d by p o l a r i s a t i o n m o d u l a t i o n i n f r a r e d d i c h r o i s m - a t e c h n i q u e s e n s i t i v e to v e r y small a n i s o t r o p i e s [34]. T h e s h o r t c h a i n s w i t h M < M e can be c o n s i d e r e d as b e h a v i n g like Rouse chains w i t h r e l a x a t i o n t i m e s on t h e order of the e x p e r i m e n t a l time-scale. F o r example, t h e Rouse t i m e s of P S D 3 a n d of P S D 1 0 a r e a p p r o x i m a t e l y 9s a n d 19s seconds, respectively at 115~ [35,36]. C o n s e q u e n t l y , t h e r e s i d u a l o r i e n t a t i o n at long t i m e s for t h e s e c h a i n s c a n be a t t r i b u t e d to o r i e n t a t i o n a l c o u p l i n g i n t e r a c t i o n s w i t h the long c h a i n s of t h e p o l y m e r m a t r i x . S i m i l a r o r i e n t a t i o n a l correlations h a v e b e e n o b s e r v e d on v a r i o u s s y s t e m s by 1H N M R s p e c t r o s c o p y s t u d i e s of s t r e t c h e d e l a s t o m e r s w h e r e even dissolved s o l v e n t m o l e c u l e s a n d free c h a i n s w e r e s h o w n to p o s s e s s a very s h o r t - l e n g t h scale local o r i e n t a t i o n [37]. T h e coupling coefficient, E, proposed by Doi e t al. [38] h a s been e x t r a c t e d from t h e o r i e n t a t i o n d a t a , y i e l d i n g a v a l u e of c a . 0.26 [36]. H o w e v e r , t h e u n c e r t a i n t y in the o r i e n t a t i o n m e a s u r e m e n t s at long t i m e s does not allow t h e

57

d e t e r m i n a t i o n of a molecular weight dependence of E, as observed by Ylitalo et al. [39] in a study of polybudadiene b i n a r y blends.

T h e cooperative n a t u r e of the o r i e n t a t i o n a l r e l a x a t i o n in a s y m m e t r i c b i n a r y blends has been f u r t h e r investigated t h r o u g h the analysis of the effect of the s h o r t chain concentration, for a given short chain length, Mw = 10 000. M a s t e r curves of the o r i e n t a t i o n a l r e l a x a t i o n of the long c h a i n s a r e p r e s e n t e d in F i g u r e 18. The long c h a i n r e l a x a t i o n in the p u r e p o l y m e r is c o m p a r e d to t h a t in the blends which c o n t a i n i n c r e a s i n g c o n c e n t r a t i o n s of s h o r t chains.

0.30

. . . .

I " " ' ' 1

. . . .

I ' ' ' ' 1 ' ' ' ' 1

. . . .

I . . . . _

O \

^

0.20

-

r

VS. OX

~

o

v

0.10

-

\

_

_ o',~,,o o

'~"

-

"otb " O ~ 9 m--.

0.00

.... -1

I .... 0

o o ~

0--10 ~ 0

.

~-a-

Dq~- O

I .... ~.... I .... i .... I .... 1 2 3 4 5 6

log t (Tg+9~ F i g u r e 18. M a s t e r curves of the orientational relaxation of t h e long chains, at Tg+9~ for PS2000 (m), PS2000/PSD10(20 wt.-%) (o) and PS2000/PSD10(30 wt.-%) (o). In a d d i t i o n to t h e u s u a l c h a r a c t e r i s t i c r a p i d decrease in o r i e n t a t i o n at short t i m e s before reaching a p l a t e a u at long times, a decrease in orientation is observed as the concentration of short PSD10 chains increases to 30%. Such a high concentration of short chains is required to affect notably the relaxation of the long species. Indeed, Figure 19 shows a negligible difference b e t w e e n PS 2000 a n d blends containing 20 wt.-% of short chains, even of very low molecular weights. O n e c o n v e n i e n t s t r a t e g y to i n t e r p r e t t h e s e r e s u l t s is to r e v i e w t h e m o l e c u l a r c h a r a c t e r i s t i c s of b i n a r y b l e n d s as e x t r a c t e d from p o l y m e r m e l t rheology [40]. The influence of short chains (M < M e) is to effectively decrease the p l a t e a u m o d u l u s and the t e r m i n a l r e l a x a t i o n times as c o m p a r e d to the p u r e p o l y m e r . C o n s e q u e n t l y , the m o l e c u l a r w e i g h t b e t w e e n e n t a n g l e m e n t s 0

0

which is related to G N by G N - pRT/Me is increased. The short chains, in other words, cause a typical diluent effect a n d t h u s the " t e m p o r a r y e n t a n g l e m e n t n e t w o r k " is looser. Therefore, a lower o r i e n t a t i o n of long chains is expected.

58 However, an i n c r e a s e in M e also implies an increase in I:A, as given by equation (3) and, therefore, the relaxation kinetics should be slower.

0.30

^

0.20

~~

0 e,i

a, V

~

9

9

0.10

o o 5"-.~

0.00 -1

0

l

2

3

4

5

6

log t (Tg+9~ Figure 19. M a s t e r curves of the orientational relaxation of the long chains at T= Tg+ 9 ~ PS2000 (0), PS2000/PSD18(20 wt.-%) (o) and PS2000/PSD3(20 wt.-%) (0). Besides Me, the relaxation time XA is proportional to a monomeric friction coefficient, ~0. This p a r a m e t e r , w h i c h describes the frictional d r a g per m o n o m e r unit as it moves t h r o u g h its environment, depends on t e m p e r a t u r e , p r e s s u r e and molecular weight (for low molecular weights) [8]. Rheological studies of the u n d i l u t e d p o l y m e r (PS2000) and the blend (PS2000/PS10) containing 30 wt.-% short chains have confirmed a decrease in the plateau modulus with the addition of the short chains. M a s t e r curves of the storage G'(co) a n d loss G"(co) moduli of the pure polymer a n d the blend are p r e s e n t e d in F i g u r e 20. The moduli were m e a s u r e d at t e m p e r a t u r e s r a n g i n g from Tg + 100 to Tg + 30~ and data were reduced to a reference t e m p e r a t u r e of 165~ The horizontal shift factors, aT, were used to e v a l u a t e the W L F coefficients C1 a n d C2 [8]. No a p p a r e n t v a r i a t i o n in the product C1C2 was detected on dilution with the short chains. In fact, an average product of 650 + 30 was obtained for both the p u r e p o l y m e r and the blend, consistent with various other reported values [41,42]. A p a r t from a reduction in the w i d t h of the p l a t e a u region and a lower plateau modulus , (G~N blend = 0.5 G 0N pure polymer), the blend exhibits a shift in the G'/G" crossover m o d u l u s a n d f r e q u e n c y in the t r a n s i t i o n zone. G oN corresponds to the value of G'(co) at the m i n i m u m of tan 5 = G"(c0)/G'(o~).

59

106

105

f

lO 4

103

........' ........i ........l ........, ........, ........l ......... ........i ........I 10.4 10-2 100 102 104 T

Figure 20. M a s t e r curves of G'(o)) (heavy lines), G"(co) at T = 165~ for PS2000 (full lines) and PS2000/PS10(30 wt.-%) (dotted lines). Due to difficulties in m e a s u r i n g the z e r o - s h e a r viscosity of such high m o l e c u l a r w e i g h t p o l y m e r s , a n d t h u s d e d u c i n g t h e m o n o m e r i c friction coefficient from G r a e s s l e y ' s ' u n c o r r e l a t e d d r a g model' [43], the following equation a d a p t e d from the modified Rouse theory has been applied [8]. G'(o~) = G"(oJ)= (appNA/4Mo)(~kBT/3)Y2o) 1/2

(11)

In e q u a t i o n 11 "ap" is a characteristic length (taken to be 7.4 .s [8]), p the d e n s i t y , NA Avogadro's n u m b e r , Mo the m o n o m e r m o l e c u l a r weight, k B B o l t z m a n n ' s c o n s t a n t , T the a b s o l u t e t e m p e r a t u r e a n d co the a n g u l a r frequency. Although this equation is based on the b e h a v i o u r of low molecular weight, u n e n t a n g l e d polymers, only the difference, r a t h e r t h a n the absolute v a l u e of friction coefficients b e t w e e n the pure p o l y m e r and the blend, is significant. F u r t h e r m o r e , the value of ~o e x t r a c t e d for the pure polymer is c o m p a r a b l e w i t h t h a t d e t e r m i n e d by e n t i r e l y i n d e p e n d e n t e s t i m a t e s from forward-recoil spectroscopy m e a s u r e m e n t s [44]. The r e s u l t s have indicated t h a t the monomeric friction coefficient is s e n s i t i v e to the p r e s e n c e of the short chains, and is smaller, at a given t e m p e r a t u r e , t h a n the homopolymer. This observed reduction in ~o h a s been shown to be only a t t r i b u t e d to an increase in the fractional free volume. Indeed, the t e m p e r a t u r e dependence of the monomeric friction coefficient is described by a WLF-type equation : lo ~~W)._ =- CI(T-To) ~To C2 + (T-To) This equation can be rewritten as :

(12)

60 CzC2 _ log~o(T) - C g + 12.303 fw log~o(T) = log~o(Tg)- Cg + (T-T=)-

(13)

where T~ = Tg-C~ is the Vogel t e m p e r a t u r e , t h a t is, the t e m p e r a t u r e at which the free volume vanishes. As mentioned above, the product C1C2 is the same for both samples. Consequently, plots of log ~0 as a function of 1/(T-T~) present identical slopes for the homopolymer a n d the blends. In fact, as shown in F i g u r e 21, not only are the slopes the same, but the intercept does not vary. Thus, the change in log ~0 upon dilution with shorter chains is only controlled by T~. A further step in the analysis of the orientation curves is to consider the role of ~0 in the relaxation processes t a k i n g into account the rheological m e a s u r e m e n t s . An alternative method of presenting the orientation-relaxation curves is to compare the pure polymer and the blend at t e m p e r a t u r e s at which ~o is the same (115~ for the homopolymer and l l 0 ~ for the PS2000/PSD10 (30 %) blend). It is shown in Figure 22 t h a t the relaxation curves of both systems can be superimposed at short times but t h a t increased deviations a p p e a r at longer times. This result suggests t h a t at short times (or short distances) the monomeric friction coefficient governs the local relaxation processes, while at longer times the relaxation dynamics depends on the environment. Recalling the Rouse and tube models, this explanation fits in with the well-established notion t h a t the local relaxation processes can be described by Rouse modes [35]. F u r t h e r m o r e , at a fixed ~o' the short time relaxational kinetics of long chains are essentially insensitive to the presence of the short species. ~,

, , , i , , , , i , , , , i , , , , i , , V , l V , , , l ~ , , ,

i

5 E Z

[]

7

~

D

g

8

[]

9

9 -10

B

F.

~e~ e

L ~ , , l

6

D

....

7

i ....

8

z ....

i ....

~ ....

9 10 103 ( 1/T-T

11 )

~ .... 12 13

oo

Figure 21. Logarithmic plot of the monomeric friction coefficient as a function of 1/(T-Too) for PS2000 (0) and PS2000/PS10(30 wt.-%) (~). It can thus be deduced t h a t the "dilution effect" of short chains is a dynamical process which does not act at short times 9 On the contrary, at longer times, as reptation theory postulates, the topological constraints which govern the chain d y n a m i c s a p p e a r looser as the concentration of short chains is increased. Indeed, even in an environment of equivalent local chain characteristics, as

61

fixed by a given m o n o m e r i c friction coefficient, the PS2000/PSD10(30 wt.-%) blend, w i t h a lower n u m b e r of effective constraints, t h a t is, a higher Me, has t h u s an i n c r e a s e d chain mobility, and, consequently, a reduced orientation.

0.30 . . . . I . . . . I . . . . , . . . . , . . . . , . . . . , ' ' ' 0.25 ^

0.20

~D

o

0.15

g~ , . O

a," 0.10 V

9 O

o

00

9

c~ 9 .~o 9 LT- I~9

2

lie ~ 1 7 6 1o 7 6

0.05

D oc:b ~

0.00

, , , , I , , ~ , l , J , , i , , , , I , , , , l , , , , l

-1

0

1

2

3

4

~o

....

5

log t (~o = Constant) F i g u r e 22. Long chain orientational relaxation reduced to reference t e m p e r a t u r e s at which log ~o = -3.54 N.s.m-l: PS2000 (o), PS2000/PSD10(30 wt.-%) (g).

8. C o n c l u s i o n

I n f r a r e d d i c h r o i s m has been successfully applied to c h a r a c t e r i z e the o r i e n t a t i o n a l r e l a x a t i o n of linear a n d b r a n c h e d polystyrene chains as well as b i n a r y blends of long and short chains. By d e u t e r a t i n g some chains or p a r t s of chains, i n f r a r e d spectroscopy provides a m e t h o d of analyzing t h e orientational b e h a v i o u r of t h e d i f f e r e n t species a n d c o n s e q u e n t l y probe t h e m o l e c u l a r relaxation mechanisms. In t h e case of l i n e a r polymers, the o r i e n t a t i o n of a chain s e g m e n t has been s h o w n to depend on its location along the chain. A more rapid orientation of chain ends has been evidenced. S i m i l a r l y , in the case of s t a r polymers, t h e r e l a x a t i o n s of the b r a n c h point, a r m centre and c h a i n end have been differentiated. The influence of b r a n c h i n g h a s been detected by c o m p a r i n g l i n e a r chains w i t h a r m s of star polymers. A slower relaxation is observed in the case of b r a n c h e d species. T h e s p e c t r o s c o p i c d a t a h a v e been c o m p a r e d w i t h t h e t h e o r e t i c a l predictions of the Doi- E d w a r d s model. In the time scale of our experiments, a q u a n t i t a t i v e a g r e e m e n t b e t w e e n e x p e r i m e n t and theory is o b t a i n e d if chain l e n g t h f l u c t u a t i o n s , r e t r a c t i o n a n d r e p t a t i o n are t a k e n into account. In the case of s t a r polymers, t h e large scale fluctuation m e c h a n i s m as proposed by P e a r s o n a n d Helfand associated w i t h the r e t r a c t i o n process is accounting for

52 the observed relaxation, except in the case of the branch point of the star, where a more rapid relaxation is experimentally observed. The infrared analyses of binary polystyrene blends have been discussed in the light of rheological measurements of the same systems. It is shown that, in addition to the existence of an orientational coupling which is essentially evidenced by the high orientation of short species, the behaviour of the long chains is sensitive at short times to the average friction coefficient in the blend. Addition of short species leads to a decrease of the friction coefficient and thus a decrease in the relaxation times. At longer times, topological effects (changes in the entanglement spacing) are superimposed to the change in the relaxation times. RE~~CES 1. See for instance Chapter 1.5 and references cited therein 2. I. M. Ward, Structure and properties of Oriented Polymers, Applied Science Pub, London, 1975 3. H. Janeschitz-Kriegl, "Polymer Melt Rheology and Flow Birefringence", Springer Verlag, Berlin, 1983 4. G. G. Ffiller, Ann. Rev. Fluid Mech., 22, 1990, 387 5. B. Jasse, J. L. Koenig, J. Macromol. Sci., Rev. Macro. Chem., C17, 1979, 61; J. Polym. Sci., Polym. Phys. Ed., 17,1979,799 6. J. F. Tassin, L. Monnerie, Macromolecules, 21, 1988, 1846 7. R. Fajolle, J. F. Tassin, P. Sergot, C. Pambrun, L. Monnerie, Polymer, 24, 1983, 379 8. J. D. Ferry, "Viscoelastic Properties of Polymers", 3rd edition, Wiley, New York 1980 9. M. Doi, S. F. Edwards, "The Theory of Polymer Dynamics", Oxford University Press, New York 1986 10. M. Doi, J. Polym. Sci., Polym. Phys. Ed., 18, 1980, 1005 11. M. Doi, J. Polym. Sci., Polym. Lett. Ed., 19, 1981, 265; J. Polym. Phys. Polym. Phys. Ed., 21, 1983, 667 12. D. S. Pearson, E. Helfand, Macromolecules, 17, 1984, 888 13. J. L. Viovy, Polymer Motions in Dense Systems, Springer Proceedings in Physics, 29, 1988,203 14. P. G. de Gennes, J. Phys. Fr., 36, 1975, 1199 15. R. C. Ball, T.C.B. McLeish, Macromolecules, 22, 1989,1911 16. J. L. Viovy, L. Monnerie, J. F. Tassin, J. Polym. Phys. Polym. Phys. Ed., 21, 1983, 2427 17. W. W. Graessley, Adv. Polym. Sci., 47, 1982, 67 18. J. Klein, Macromolecules, 19, 1986, 105 19. M. Doi, W. W. Graessley, D. S. Pearson, E. Helfand, Macromolecules, 20, 1987, 1900 20. M. Rubinstein, E. Helfand, D. S. Pearson, Macromolecules, 20, 1987, 822 21. M. Rubinstein, R. H. Colby, J. Chem. Phys., 89, 1988, 5291 22. A. Schausberger, H. Knoglinger, H. Janeschitz Kriegl, Rheol. Acta, 26, 1987, 4668 23. J. F. Tassin, L. Monnerie, J. Polym. Sci., Polym. Phys. Ed., 21, 1983, 1981 24. J. A. Kornfield, G. G. Fuller, D. S. Pearson, Macromolecules, 22, 1989, 1334

63 25. G. G. Fuller, C. M. Ylitalo, J. of non crystalline Solids, 131, 1991, 676 26. P. F. Green, P. J. Mills, C. J. Palmstrom, J. W. Mayer, E. J. Kramer, Phys. Rev. Lett., 53, 1984, 2143 27. J. F. Tassin, Th~se de doctorat d'Etat, Universit~ Paris VI, 1986 28. L. R. G. Treloar, "The Physics of Rubber Elasticity", Oxford Univ. Press. London, 1958 29. J. F. Tassin, L. J. Fetters, L. Monnerie, Macromolecules, 21, 1988, 2404 30. C. M. Ylitalo, G. G. Fuller, V. Abetz, R. Stadler, D. S. Pearson, Rheol. Acta, 29, 1990, 543 31. A. Lee, R. P. Wool, Macromolecules, 20, 1987, 1924 32. K. Osaki, E. Takatori, M. Ueda, M. Kurata, T. Kotaka, H. Ohnuma, Macromolecules, 22, 1989, 2457 33. W. J. Walczak, R. P. Wool, Macromolecules, 24, 1991, 4657 34. T. Buffeteau, B. Desbat, S. Besbes, M. Nafati, L. Bokobza, Polymer, 35, 1994, 2538 35. P. J. Rouse, J. Chem. Phys., 21,1953, 1272 36. J. F. Tassin, A. Baschwitz, J. Y. Moise, L. Monnerie, Macromolecules, 23, 1990, 1879 37. H. Toriumi, B. Deloche, J. Herz, E. T. Samulski, Macromolecules, 18,1989, 1488 ; B. Deloche, A. Dubault, J. Herz, A. Lapp, Europhys. Lett., 1,1986, 629 38. M. Doi, D. Pearson, J. Kornfield, G. Fuller, Macromolecules, 22,1989, 1488 39. C. M. Ylitalo, J. A. Zawada, G. G. Fuller, V. Abetz, R. Stadler, Polymer, 33,1992, 2949 40.J.P. Montfort, G. Marin, P. Monge, Macromolecules, 17,1984, 1551; 19,1986, 1979 41. D. J. Plazek, J. Phys. Chem., 69,1965, 3480 42. P. Lomellini, Polymer, 33,1992, 4983 43. W. W. Graessley, J. Chem. Phys., 54,1971, 5143 44. R. J. Composto, E. J. Kramer, D. M. White, Polymer, 31, 1990, 2320


Rheology for Polymer Melt Processing J-M. Piau and J-F. Agassant (editors) 9 1996 Elsevier Science B.V. All rights reserved.

65

Chain conformation in elongational and shear flow as seen by SANS R. Muller and C. Picot Institut Charles Sadron (CRM-EAHP), 4-6 rue Boussingault, 67000 Strasbourg, France.

1. INTRODUCTION The central problem in the understanding of the rheological properties of polymer melts is to achieve a description at a molecular level of such highly entangled and consequently interacting long flexible chains systems. This goal is clearly of fundamental interest as well as of practical importance with regard to polymer processing. A variety of molecular models have been proposed to represent the behaviour of polymer chains in interaction with their surroundings. Extensive reviews describing the molecular approaches of polymer viscoelasticity can be found in monographs discussing the limits of their applicability [ 1,2]. In these models, equilibrium and dynamical properties are described in terms of spatial distribution of the chain segments. Whatever the models invoked, from the simplest (i.e the Rouse model) to the more sophisticated (reptation model,...), the stress is directly related to the orientation of the bound vectors of the polymer chain. Therefore a direct knowledge of the spatial distribution of the chain segments is the key point in the molecular description of the macroscopic viscoelastic behaviour. Among all experimental techniques, radiation scattering is the most appropriate to accede to this topological description. The scattering is indeed directly related through Fourier transform with the spatial correlation of fluctuations. More specifically, when observing a polymer system containing labelled chains, the distribution of chain segments in space can be analyzed [3]. With regard to polymeric condensed systems, small angle neutron scattering (SANS) is particularly well suited for studying chain conformations [4]. The advantages of SANS are twofold: the available range of scattering vectors on low angle spectrometers (10-3 - 1 A"t) matches particularly well the polymer chain dimensions. neutron scattering is the result of neutron-nucleus interaction and hence different isotopic species can interact with neutron differently. In particular, hydrogen and deuterium have coherent scattering lengths (arr=-0.374x 10~2; aD=+0.667x 10~2 cm respectively) which differ in both sign and magnitude. Consequently, a deuterated molecule in a protonated matrix will be "visible" by neutrons and vice versa. The SANS technique has already contributed to the analysis of polymer conformation in systems at equilibrium (i.e. dilute and concentrated solutions, bulk) [5-7] and to the molecular response of polymer systems under external constraints (i.e. polymer networks, polymer melts under uniaxial or shear deformation) [8-14]. The aim of the present work has been to establish correlations between bulk macroscopic response of polymer melts under flow and the behaviour at a molecular level as seen by SANS, and to discuss the results in the frame of molecular theories. Two simple and well defined geometries of deformation have been investigated: uniaxial elongation and simple shear. The -

-

66 basic experiment consists in submitting the polymer melts containing deuterium labelled chains to different, well-defined regimes of flows (transient and steady flows, relaxation, creep) and then perform SANS measurements on probes where the chains have been frozen in by rapid quenching below the glass transition temperature Tg.

2. METHODOLOGY 2.1. Polymers used All samples consist of anionically synthesized PS containing deuterated polystyrene (PSD) chains. Two hydrogenated (PSH1 and PSH2) and two deuterated (PSD 1 and PSD2) polymers were synthesized. Their weight-average molecular weight was determined by light-scattering measurements in benzene taking into account the refractive index increments for PSH and PSD. The polydispersity was characterized by gel permeation chromatography using the universal calibration. As shown in Table 1, PSH2 and PSD2 have a weight-average polymerization index which is about ten times that of PSH1 and PSD 1.

Table 1 Molecular WeiBht Characteristics of Polymers Polymer

Ms

Mw

Mz

M,dM,

Il~

PSH1 PSD 1 PSH2 PSD2

80 000 81 000 865 000 975 000

90 000 95 000 970 000 1 170 000

99 000 108 000 1 057 000 1 355 000

1.12 1.13 1.12 1.2

865 848 9 330 10 450

With these polymers, two narrow MWD samples (S 1 and $2) and seven binary blends were prepared for the SANS experiments. The composition of all samples is given in Table 2. They were obtained by freeze-drying a solution of all constituents in benzene [9]. For the narrow MWD samples, the weight fraction ~D of deuterated species is 10%. The weight fraction O2 of high molecular species in the binary blends ranges from 10% to 60%; they contain either PSD 1 03'20 and B'40) or PSD2 chains (B 10, B20, B40 and Br0), which allows one to investigate respectively the conformation of the low or high molecular weight species. As seen in Table 2, the total weight fraction ~ of deuterated species is lower for the binary blends than for the narrow M W samples. The concentrations were calculated so that the measured scattering intensity remains simply related to the form factor of the deuterated species (see section 2.3.). The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140~ for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature.

67

Table 2 Composition in weight fractions of all samples Sample

~PSm

~PSD~

q~Psm

(~)PSD2

~2

S1 $2

0.9 0

0.1 0

0 0.9

0 0.1

0 1

0.1 0.1

B10 B20 B40 B60

0.9 0.8 0.6 0.4

0 0 0 0

0.092 0.189 0.382 0.574

0.008 0.011 0.018 0.026

0.1 0.2 0.4 0.6

0.008 0.011 0.018 0.026

B'20 B'60

0.76 0.36

0.04 0.04

0.2 0.6

0 0

0.2 0.6

0.04 0.04

n

Figure 1. Relaxation spectra at 140~ of samples S 1, $2, B 10, B20 and B40.

--Im-- $1

v

BIO - - ~ - - B20 - - V - - B40 - - ~ - - $2

o

-2

-1

0

1

2

3

4

5

log(x) (s)

Table 3 Linear viscoe!astic parameters at 140~ of all samples. Samp!e

110 (106 Pa.s)

Jr ~ (10 s Pa ~)

X.(s)

S1 B10 B20, B'20 B40 B60, B'60 $2

0.65 9 42 290 780 3550

1 60 27 7.8 4.3 1.6

6.5 5 400 11 300 22 600 33 500 56 800

68

2.2. Experimental setup The data in this work were obtained by quenching either uniaxially stretched samples for which the low thickness of the specimens allows rapid cooling, or samples deformed in simple shear where higher thickness of the specimens was required in order to perform the scattering experiments along the three principal shear directions. For both types of flow, special devices were developed to control the flow kinematics in the molten state as well as the quenching process which freezes the molecular orientation. These devices are briefly described in the next paragraphs. 2.2.1. Eiongational flow For the elongational flow experiments, all specimens were uniaxially stretched with the extensional rheometer schematically shown in Fig. 2. A detailed description of this instrument has been published elsewhere [ 15]. Its main features are the following: (a) symmetric stretching with respect to the centre of the specimen (this allows measurement of flow birefringence at a fixed point of the sample), (b) temperature control with a double silicon-oil bath, which can be removed by vertical displacement. This oven provides a good temperature control (less than • ~ gradient in the whole bath), whereas buoyancy prevents the specimen from flowing under the effect of gravity. Homogeneous deformation is thus achieved even at very low strain rates for which the time that the specimen stays in the molten state is longer than the terminal relaxation time of the polymer sample. Stretched specimens are quenched by rapid removal of the surrounding oil bath.

Figure 2. Schematic drawing of extensional rheometer. (a)motors, (b)screws, (c) transducer, (d) sample, (e) oil bath, (f) laser, (g) lower position of bath.

69 2.2.2. Shear flow

The experimental device constructed to orient uniformly thick samples in simple shear is schematically represented in Fig. 3. It is basically a sliding-plate rheometer, the polymer sample being sheared between two temperature-controlled parallel plates. The upper plate is fixed whereas the lower plate can be displaced both horizontally and vertically with two pneumatic jacks. The shearing experiment occurs in the following way: (a) the temperature of both plates is set to its value during the melt flow by a silicon oil circulation before the polymer sample is inserted between the plates; (b) once thermal equilibration of the sarriple is achieved, the gap between the plates is fixed by a mechanical stop; (c) a constant shear force Fs is applied to the melt with the horizontal jack during a given time. At the same time, the normal force Fr~ in the sample is recorded with a force transducer mounted below the lower plate. From the horizontal displacement, the shear strain is obtained as a function of time; (d) after shearing, the sample is cooled by stopping the oil circulation and starting the cold water circulation which is located close to the surface of the plates. Throughout the cooling stage, both the shear force Fs and the normal force F~ are kept constant, the vertical jack being now driven with the same compressive force which has been measured during the sheafing stage. Keeping a constant compressive force on the sample during cooling allows compensation for thermal shrinkage and prevents the polymer from slipping away from the surfaces of the rheometer. Furthermore, if the temperature is assumed to be uniform in each polymer layer (at constant height) at any time during cooling, each polymer layer will experience the same constant shear and normal stresses during quenching. This in turn means that if steady flow is reached at the beginning of the cooling stage, the orientation will be uniform throughout the whole thickness of the sample, although the inner layers, which will be cooled later, will experience a higher final shear strain than the outer layers in contact with the plates which are quenched first. Birefringence provides an easy way to check the uniformity of orientation in the sample thickness: the observation with a polarizing microscope of small strips cut out vertically in the x-y sheafing plane shows that both the.birefringence and the extinction angle are uniform in the whole thickness (see also chapter III.1). Further experimental details have been previously published [ 13].

Figure 3. Schematic drawing of shear apparatus: (a) specimen, (b) oil circulation, (c) water circulation, (d) normal force jack, (e) shear force jack. x, y and z are the principal shear directions.

70

2.3. Small-angle neutron scattering (SANS) in bulk polymers 2.3.1. Theoretical background Detailed reviews about application of SANS to the study of polymer in bulk are available in the literature [3]. These reviews provide information on the experimental technique as well on as on the fundamental theories related to scattering by polymers. Here, we will restrict ourselves to the description of some basic concepts to enable a better understanding of the experimental results that will be presented. A schematic representation of an experimental device used for SANS measurements is shown in Fig. 4.

I I

x

I

Stretching direction %

z Neutrons

IIIII ~

I \\\x~,,X

t

q Sample

Y

9

I

Y

Figure 4. Schematic representation of an experimental setup for SANS.

As mentioned before, the scattered intensity arises from the interaction between neutrons and nuclei. It decomposes into two terms, namely the coherent and the incoherent contribution. The coherent intensity depends on the scattering vector q which is the difference between the momentum of the incident and the scattered neutrons. Its norm is q=(4r~/k) sin(0/2), 0 being the

71 scattering angle and ~, the neutron wavelength. The incoherent intensity arises mainly from H nuclei and is not q dependent in the small q range of SANS. Its contribution results in a fiat background which can be taken into account by appropriate subtraction. For a mixture of identically D-labelled chains in a matrix of H chains with the same polymerization index, the coherent scattered intensity is given by:

S(q)= N AP~2-2(aD - all)2~D(I- q)D)P(q)=KM~D (1- q)D)P(q)

(1)

where aD and aH are the coherent scattering lengths of D and H monomer units, m and M are the molar weights of monomer unit and polymer chain, p is the density, ~D is the volume fraction of labelled chains, NA is Avogadro's number and P(q) the intra chain correlation function relative to the labelled coil: P(q) = ~

1

(2)

Z (exp(iq. rkl))

k,1

where r~a is the vector joining subunits k and 1 of the chain formed by N segments, < > is an ensemble average over all chain conformations in the sample.

Isotropic samples. For an isotropic set of identical coils, the form factor only depends on 0 and is given by the Debye function g(x):

g(x) = ~--

with x=q2xRg2, Rg being the radius of gyration of the chains. In the Guinier range of q ( qR~COS2 X 0 + < X l I 2 >sin2 X0 - 2 <XlXii > s i n x 0 cosg0

(24)

y2 > = < x i 2 >sin 2 Xo+<Xii 2 >cos 2 go +2 <xixii > s i n x o cosgo If the principal directions for the chain orientation were X 0 and 7~0+rt/2 at all scales, the mean value of the product XlXn would be zero (due to the symmetry of the conformation distribution with respect to the I and II axes). In this case, the following relations between the mean square chain dimensions would hold: R g,x = Rgj 2 COS2 X0 + Rg,II 2 sin2 X0 2 = Rgj 2 si n 2 Zo + RgjI 2 cOs2 X0 Rg,y

(25)

In Table 7, the experimental values ofR~. ,,x and Rg,y determined in the x-z and y-z shearing planes are compared to the values calculate~l from Rg,I, Rg,i I and g 0 according to Eq. 25. The observed agreement indicates that the q-dependence of ~ at local scales as shown in Fig. 29 has little influence on the mean square chain dimensions Rg,i and Rg, II. As a matter of fact, these quantities are determined in the low q-range and involve mainly large scale position correlations for which the principal directions of orientation are equal to ~ 0 and X 0+zc/2-

92 In section 3.1.3. we proposed a simple model to calculate the anisotropic form factor of the chains in a uniaxially deformed polymer melt. The only parameters are the deformation ratio Z, of the entanglement network (which was assumed to be identical to the macroscopic recoverable strain) and the number ne of entanglements per chain. For a chain with dangling end submolecules the mean square dimension in a principal direction of orientation is then given by Eq. 19. As seen in section 3.1.3. for low stress levels n can be estimated from the plateau modulus and the molecular weight of the chain (n~-5 por polymer S 1). The molecular deformation ratio Z, in the directions I and II can be estimated in the following way: the difference Ao between the principal stresses in the x-y plane can be readily calculated from the birefringence A (measured parallel and perpendicular to the direction of extinction) and the stress-optical coefficient C for molten polystyrene (C = 4.8x10"gPa"1, see Chapter III. 1). According to the classical network theory, the stress tensor is proportional to the Cauchy deformation tensor which means that the network deformation along the principal directions of the stress tensor are Z, and I/Z, where:

k2 __}_.1 = Ao

(26)

The values of Z, calculated from Eq. 26 are given in Table 8. It should be pointed out that the classical rubber theory does not account for a dependence of molecular orientation on the considered length scale, as found for sample C in Fig. 29. Therefore the simplistic approach presented here should in principle be restricted to samples A and B for which the principal directions for the low-q SANS data (I and II) are close to the principal directions for the refractive index. The data in Table 8 actually show a satisfactory agreement between calculated and experimental mean square chain dimensions for samples A and B. For sample C, the agreement is less satisfactory in the direction perpendicular to the chain elongation, but one must be aware that for this sample, the directions in which Rg I and Rg, i I have been measured are different from those corresponding to the calculation of Eq. 26.

Table 8 Extension ratio and root mean square dimensions in the principal directions of stress calculated from the network model (see text). . ..... Sample

A

B

. . . . .

C

A XA AO (Pa) Z, l/Z,

9.4x 10.4 26~ 2• 105 1.27 0.79

1.08x 10-3 24~ 2.3• 105 1.32 0.76

3.8x 10-3 22"5~ 8.1• 105 2.07 0.48

Rg(~) (A) I/Z,) (A) ,I (~) ,II (A)

98 70 94 73

101 69 99 69

148 56 143 47

93

5. CONCLUSIONS AND PERSPECTIVES The reliability of the experimental procedure (deformation in the melt, quenching, characterization of orientation at room temperature) could be verified in elongation (specimens quenched for different macroscopic extensions in steady elongational flow) as well as in simple shear (good correlation of the chain dimensions measured in all principal shear directions). In the uniaxial elongational flow geometry, the proposed model of a temporary network leads to a satisfactory description of the SANS data both in the intermediate scattering vector range and in the Guinier range. The number of elastic strands per chain determined from both stress and SANS data is approximately the same. The agreement also holds for high molecular weight chains and binary blends. For these systems, SANS proves to be a powerful technique since selective deuteration allows one to characterize the orientation of the various components of the blends. For the characterization of chain orientation in shear, specific experimental problems had to be solved. An apparatus was specifically designed for the purpose of quenching thick specimens oriented in simple shear with good homogeneity of orientation throughout the thickness. The data obtained with the scattering vector in the x-y principal shearing plane showed that the principal directions of orientation depend on the magnitude of the scattering vector, and therefore on the considered length scale on the chain. On the other hand, the position correlations within the chain in the neutral direction of the flow have been found to remain unaffected by the shear deformation. It should be mentionned that the result of q-dependent chain orientation in shear was also found by nonequilibrium molecular dynamics (NEMD) simulations by Kr6ger et al [32]. The increasing power of computational techniques will surely result in increased accuracy and usefulness of this type of numerical simulation.

REFERENCES

.

.

.

9. 10. 11.

Doi M. and Edwards S.F., The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. Bird R. B., Armstrong R.C. and Hassager O., Dynamics of Polymeric Liquids, Wiley, New York, 1987. Higgins J.S. and Benoit H., Polymer and Neutron Scattering, Oxford Science Publications, 1994. Picot C., Static and Dynamic Properties of the Polymeric Solid State, NATO Advanced Study Series, Vol.94, Pethrick A. and Richards R.W. (Eds), Reidel Publishing. Cotton J.P., Decker D., Benoit H., Farnoux B., Higgins J., Jannink G., Ober R., Picot C. and des Cloizeaux J., Macromolecules, 7 (1975) 863. Kirste R.G., Kruse W.A. and Ibel K., Polymer, 16 (1975) 20. Daoud M., Cotton J.P., Famoux B., Jannink G., Sarma G., Benoit H., Duplessix R., Picot C. and de Gennes P.G., Macromolecules, 8 (1975) 804. Picot, C., Prog. Colloid Polym. Sci., 75 (1987) 83. Muller R., Picot C., Zang Y.H. and Froelich D., Macromolecules 23 (1990) 2577. Bou6 F., Bastide J. and Buzier M., Springer Proc. Phys., 42 (1988) 65. Lindner P and OberthOr R., Colloid Polym. Sci., 263 (1985) 443.

94 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 28. 29. 30. 31.

Lindner P. and Oberth0r R., Physica B, 156 (1989) 410. Muller R., Pesce J.J. arid Picot C., Macromolecules, 26 (1993) 4356. Bastide J., Herz J. and Bou6 F., J. Phys. (Les Ulis, Ft.) 46 (1985) 1967. Muller, R; and Froelich D., Polymer, 26 (1985) 1477. De Cremes P.G., Scaling Concepts in Polymer Physics, Comell University Press, 1979. M0nstedt H., Rheol. Acta, 14 (1975) 1077. MOnstedt, H. and Laun H.M., Rheol. Acta, 18 (1979) 492. Delaby I., Ernst B., Germain Y. and Muller R., J. Rheol., 38 (1994) 1705. Rouse P.E., J. Chem. Phys., 21 (1953) 1272. Ferry J.D., Landel R.F. and Williams M.L., J. Appl. Phys., 26 (1955) 359. Rabin Y., J.Chem.Phys., 88 (1988) 4014. Green M.S. and Tobolsky A.V., J. Chem. Phys., 14 (1946) 80. Yamamoto M., J.Phys.Soc.Japan, 11 (1956)413; 12 (1957) 1148; 13 (1958) 1200. Lodge A.S., Armstrong R.C., Wagner M.H. and Winter H.H., Pure Appl. Chem., 54 (1982) 1349. Wagner M.H., Rheol. Acta, 18 (1979) 33. Muller R., Froelich D. and Zang Y.H., J. Polym. Sci., Polym. Phys. Ed., 25 (1987) 295. Bou6 F., Bastide J., Buzier M., Lapp A., Herz J. and Vilgis T.A., Colloid Polym. Sci., 269, 195 (1991). Ullman R., Macromolecules, 15 (1982) 1395. Muller, R. and Picot, C., Makromol. Chem., Macromol. Symp., 56 (1992) 107. Lindner P., Neutron, X-Ray and Light Scattering, Lindner P. and Zemb T. (Eds); North-Holland, Amsterdam, 1991. KrOger M., Loose W. and Hess S., J. Rheol., 37 (1993) 1057.


95

Molecular Rheology and Linear Viscoelasticity G. Marin and J.P. Monffort Laboratoire de Physique des Matdriaux Industriels, URA-CNRS 1494, Universit6 de Pau et des Pays de r Adour, 64000 Pau (France) 1. I N T R O D U C T I O N M e a s u r e m e n t of the linear viscoelastic properties is the basic rheological characterization of polymer melts. These properties may be evaluated in the time domain (mainly creep and relaxation experiments) or in the frequency domain: in this case we will talk about mechanical spectroscopy, where the sample experiences a harmonic stimulus (either stress or strain). One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the m a t e r i a l u n d e r study. F u r t h e r m o r e , linear viscoelasticity a n d n o n l i n e a r viscoelasticity are not different fields that would be disconnected: in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation t h a t would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. On the other side, the linear viscoelastic functions of polymer melts are directly related to their molecular structure, and these relations are now defined well enough to derive, with a reasonable accuracy, what would be the linear viscoelastic behaviour of a given species from its molecular weight distribution, at least in the case of linear polymers. Moreover, there is a definite trend to use rheology as a molecular characterization tool, in the same way as other popular spectroscopic methods such as N.M.R spectroscopy or Size Exclusion C h r o m a t o g r a p h y . However, the inverse problem, i.e., getting molecular weight distribution from rheological measurements, is a difficult (and ill-defined) problem, and this is a very up-to-date area of research due to its obvious practical implications in polymer characterization. The relationship between chemical structure and viscoelastic behaviour is established through molecular models considering that polymers relax or diffuse in the same way: they are considered as flexible statistical chains trapped between the topological constraints created by the surrounding chains.

96 2. L I N E A R VISCOELASTIC BEHAVIOUR OF L I N E A R AND FLEXIBLE CHAINS . BASICS AND P H E N O M E N O L O G Y We will begin with a brief survey of linear viscoelasticity (section 2.1) 9we will define the various material functions and the mathematical theory of linear viscoelasticity will give us the mathematical bridges which relate these functions. We will then describe the main features of the linear viscoelastic behaviour of polymer melts a n d c o n c e n t r a t e d solutions in a p u r e l y r a t i o n a l a n d phenomenological way (section 2.2)" the simple and important conclusions drawn from this analysis will give us the support for the molecular models described below (sections 3 to 6). 2.1. L i n e a r viscoelastic functions The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the frequency domain are purely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. In t h e various f o r m u l a t i o n s of the m a t h e m a t i c a l t h e o r y of l i n e a r viscoelasticity, one should differentiate clearly the m e a s u r a b l e and nonmeasurable functions, especially when it comes to modelling: a p a r t from the material functions quoted above, one may also define non-measurable viscoelastic functions which are pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and the memory function. These mathematical tools may prove to be useful in some situations: for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the differential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 2.1.1. Functions in the time domain The relaxation modulus G(t) is the value of the transient stress per unit strain in a step-strain experiment. This type of experiment may be achieved with modern rotary rheometers with a limited resolution in time (roughly 10 -2 s). If one wishes to evaluate G(t) at shorter times, it is necessary to derive G(t) from the high frequency G*(co) data by an inverse Carson-Laplace transform. The creep function J(t) is the transient strain per unit stress in a step-stress experiment. The resolution at short times is also limited from instr~ment response and sensitivity. J(t) at short times may also be derived from the high frequency complex compliance data.

97 One m a y also use a memory function m(t) within the integral formulation of l i n e a r a n d nonlinear viscoelasticity; this memory function m(t), which is t h e derivative of the relaxation function G(t), is not a measurable function.

2.1.2. Functions in the frequency domain Three complex functions m a y be used to characterize the linear viscoelastic behaviour in the frequency domain" - The complex shear modulus, which is the complex stress to complex strain ratio" G * (co) = complex stress c*(r - ------- = G'(co)+ j G"(co) y*(~) complex strain

(2-1)

- The complex viscosity, which is the complex stress to complex rate of strain ratio" 11"(o)) =

complex stress = (co-----)) ~* = Tl' (co)- j 11"(o)) = ~ _*(co) _G complex rate of strain ~/* (co) jco

(2-2)

- The complex compliance, which is the complex strain to complex stress ratio" j .(co) = complex strain = 7*(co) = j,(co)_ j j,,(co) = 1 (2-3) complex stress ~* (co) G* (co) These three functions are related via simple algebraic relations" G * (co) = ~

1

J*(r

= jco T1* (co)

(2-4)

Although the complex shear modulus is not the most appropriate function to use in all cases, we will describe the linear viscoelastic behaviour in terms of this last function, which is the most referred to experimentally; furthermore, molecular models are mostly linked to the relaxation modulus, which is the inverse Fourier transform of the complex shear modulus.

2.1.3. The distribution of relaxation times By analogy with a generalized Maxwell model, it is possible to write the relaxation modulus or the complex modulus as a sum of the contributions of n individual Maxwell models" 12

G(t)=~G k=l

12

k(t)=~G ke

----

t

~k

(2-5)

k=l 4oo

and G * (co)= jco ~ G(t) e -j~t dt 0

(2-6)

98 n

n

hence G*(co)= ~ G k *(co)= ~ jOTlk k:l k=ll+f~k

(2-7)

with Tlk=Gk~k. In terms of molecular models (section 3), the set of individual times ~k and weighting factors Gk is imposed by the model. It is also possible to derive a set of relaxation times and weighting factors numerically by optimization or approximation methods from the experimental data. In that case, the relaxation times have no real physical meaning and are simply numerical/empirical parameters which allows one to represent the viscoelastic behaviour as a sum of decaying exponentials which are handy to use for numerical analysis. It is also possible to give integral forms of these sums" oo

G * (co) = jr

H(z) 0 ~+jcoz dz +oo

(2-8)

t

and G(t)= ~ H(z)e ~ d ln(z)

(2-9)

--oo

H(z) is a continuous distribution of the logarithms of relaxation times and is called the "relaxation spectrum" by rheologists, whilst the "true" distribution of relaxation times is zH(z). We have r e p o ~ on Figure I the normalized distribution of relaxation times for 4 polystyrene samples with polydispersity indices ranging from 1.05 to 4.2 [2]. It is clear that the distribution of relaxation times broadens with the distribution of molecular weights; these features will be analyzed in terms of molecular models in sections 3 to 6.

A

v

o

I

-3

-2

-I

0 Log z / Z o

+1

*2

Figure 1 9Reduced distribution of relaxation times for atactic polystyrene samples having different values of polydispersity index P=l.05 (A), 1.25 (B), 2.45 (C), 4.2 (D)[ref. 2]

99 2.2. T h e m a i n f e a t u r e s of the l i n e a r viscoelastic b e h a v i o u r of p o l y m e r melts: We will discuss in this section the variations of the viscoelastic parameters derived from linear viscoelastic measurements; all these parameters may be derived from any type of m e a s u r e m e n t (relaxation or creep experiment, mechanical spectroscopy) performed in the relevant time or frequency domain. The discussion will be focused however on the complex shear modulus which is the basic function derived from isothermal frequency sweep measurements performed with modern rotary rheometers.

-

log G* (W)

G,.-...-..~ ~ - . - - . ~

G' ~ G

fMu

-//,,'M2 f

~~

G o'

G

t er min al

e

log UJ plat eau

POLYMERIC

"BEHAVIOUR

transition

glass y

SMALL SCALE MOLECULAR MOTIONS

Figure 2 9Schematic of the variations of the complex shear modulus of linear polymers (dotted lines: molecular weight M2>M1). The main features of the linear viscoelastic behaviour of polymeric melts in the frequency domain are reported on Figure 2 : - the lowest frequency range describes the slowest relaxation motions of the macromolecules. The double logarithmic plot of G' and G" exhibits slopes of -respectively- 2 and 1, leading to two characteristic parameters. The zero-shear viscosity" ~o = lim G"(co____~) co-,0

CO

(2-10)

100

The limiting compliance:

G'(r j0=lim ~-~o [G"(co)] 2

=1

limG'(c~

Tioz ~-,o r ~

(2-11)

which is the elastic parameter governing the main features of the elasticity of the melt (first normal stress difference, extrudate swell, etc...). The zero-shear viscosity is the norm of the relaxation spectr~m :

Tlo= ~~H(z)d In z

(2-12)

The product TIo jo is the characteristic relaxation time zo of the terminal region. In terms of molecular models, this time scales as the longest relaxation time. In terms of the distribution of relaxation times H(z), zo is the "weight-average relaxation time" which is the average relaxation time related to the second order moment of the relaxation spectm~m : ~o = no jo =< ~w >= ~ z2H(z)d In

(2-13)

At intermediate frequencies, monodisperse polymers exhibit a well-defined "plateau region" where G'= constant G~ (Figure 2). For a given macromolecular species, the value of the plateau modulus is a characteristic feature t h a t does not depend on molecular weight. The only way to lower the plateau modulus is to add small compatible molecules, either of the same species or not : this is, for example, what is done for Hot-Melt Adhesives (HMAs) when adding a "tackifying resin" which softens the polymer and improves the "tack". In terms of the distribution of relaxation times, the ratio TIo/G~ is an average relaxation time (we may call it "n-tuber-average relaxation time"), which is the first order moment of the normalized relaxation spectm, m : < ZN >= no/ G~ = i ~H(z)d In H(~)d In z

(2-14)

The ratio: < Zw >

o co = Je

(2-15)

is a "polydispersity index of relaxation times" which characterizes the broadness of the distribution of relaxation times. For monodisperse species, the experimental value of this ratio lies between 2 and 2.5, whatever the polymer nature. This value increases largely as polydispersity increases. One of the direct practical

lO1 applications of molecular modelling will be to relate the distribution of molecular weights to the distribution of relaxation times.

2.2.1. The effects of chain length 2.2.1.1. The zero-shear viscosity Tl0 The variations of the zero-shear viscosity of monodisperse polymeric melts and concentrated solutions exhibit two domains, each being characterized as a first approximation by a power law exponent" at low molecular weights, corresponding to less than 200 monomeric units, the exponent lies between 1 and 1.5. This domain will be analyzed in section 5. above a critical molecular weight Mc, the power law exponent is 3.4-3.5. As far as viscosity is concerned, Mc defines the begining of a regime where the macromolecular chains are viewed as "entangled", which explains the large molecular weight dependence of viscosity. The entangled regime will be modelled later using the "reptation" concept (see section 3). -

-

f

!

i

i 0 K)

_

/

o/+

13

vi r I0 e n

I'--I

(J

o

o

c;

.

~_

/,

/

g

/

#o,+

IO s -

o/

_

0

10 4 -

10 2 10 3

I

10 4

I

10 5

I

10 6

log M

Figure 3 : Molecular weight dependence of the structure factor of nearly monodisperse polystyrene samples [3]. Figure 3 illustrates this type of behaviour for various series of anionic polystyrene samples with a fairly narrow distribution of molecular weights. In the case of entangled polydisperse materials (Mw>>Mc), the zero-shear viscosity follows approximately the same molecular weight dependence as for monodisperse species when the viscosity data is plotted as a function of the weight-average molecular weight Mw; i.e. : 110 = A(T) M~ 4

(2-16)

I02 2.2.1.2. The plateau modulus G~ The plateau region begins to be developed at molecular weights somewhat above Mc; it is however a well-defined elastic parameter for a given chemical species of high molecular weight. Comparing different polymer species, its value increases with the flexibility of the chain (i.e: GNPolystyrene< o o GNPolyethylene). The use of the theory of rubber elasticity may give an order of magnitude of the average molecular weight Me between entanglements which create a temporary network: G~ = pRT

(2-17)

Me '

p being the polymer density and T the absolute temperature. The critical molecular weight Mc is roughly two times the molecular weight between entanglements Me.

2.2.1.3. The limiting compliance jo In the entangled regime, the limiting compliance jo of monodisperse samples is also an elasticconstant characterizing a given polymer chemical species: contrary to the plateau modulus, its value depends on the distribution of molecular weights, i.e.,on the polydispersity index as a firstapproximation. This is a very important point, as m a n y elasticeffects(firstnormal stress difference,extrudate swell, ...)of the melt are governed by the limitingcompliance. For purely monodisperse samples, the product jo G ~ has a value close to 2 for all flexiblepolymers. Below the entangled regime, the limiting compliance follows approximately a linear dependence with molecular weight according to Rouse's theory (see section 6)jo = 0.4 M . pRT

(2-18)

The molecular weight value M'c where the compliance becomes independent of molecular weight is larger than M c (M'c--3Mc), which indicates that the "polymeric" regime seems to appear at higher molecular weights for elastic properties compared with viscous properties. So one has to keep in mind that the chain length (or molecular weight) at which "entanglements" effects begin to appear depends strongly on the physical property measured (melt viscosity, melt elasticity,self-diffusion,etc...)(see in particular chapter L1). In the case of polydisperse polymers, the limiting compliance increases strongly with the broadness of the distribution of molecular weights. The limiting compliance is not, however, a simple function of the polydispersity index, because its value depends on the shape of the distributionitself.There is indeed no simple correlation with any molecular weight moments (averages), and molecular models will be really helpful to describe the elasticity of the melt.

103

2.2.2. The effect of temperature: For a given polymer, the viscoelastic curves (either moduli or compliances) obtained at different temperatures in the plateau and terminal regions are simply afflne in the frequency (or time) scale, in a double logarithmic plot. The use of this time-temperature equivalence allows one to obtain "master curves" at a reference temperature, which enlarges considerably the experimental window. For glass-forming materials such as polystyrene, polymethylmetacrylate, polycarbonate, polymerists describe the shift factor aT in terms of the WLF equation: -c~ -To) In aT = (co + T - T o ) '

(2-19)

T being the experimental temperature and To the reference temperature to which the data is shifted. The WLF may be reduced to the Vogel equation which describes the viscosity of molten glasses and supercooled liquids"

B/af

In a T = _T0 ---~-

B/af W-'-~_ '

(2-20)

where the limiting temperature Too may be related to the glass transition temperature Tg by the approximate rule: Tg-Too = c2g -- 60 ~ C. The entropic nature of the elasticity of the melt implies also a slight vertical sbJR in the plateau and terminal regions. This shi~" b T = P~176 pT'

(2-21)

may be neglected when using the time-temperature equivalence in a limited range of temperatures. The time-temperature equivalence implies that the viscosity (or relaxation times) of polymers may be written as the product of two functions : no = $(P(M)). M (T)

(2-22)

The mobility factor M (T) describes the segmental mobility of the chain : it depends mostly on temperature and pressure, but may be affected by the presence of small chains (such as solvent molecules or small chains of the same chemical species as the polymer). For concentrated polymer solutions, the addition of small molecules affects mostly the glass transition temperature (hence Too), and the value of B (eq.2-20) is essentially the same as for the bulk polymer. A plastifyer will decrease the value of Too, and hence increase the segmental mobility. On the contrary, the addition of a tackifying resin which has a higher Tg than the polymer will increase the segmental mobility of the polymer in the case of formulations of Hot-Melt adhesives.

104 The structure factor $(P(M)) describes the topological relaxation of the macromolecular chains: t ~ s is the function which will be described by molecular models, P(M) being the distribution of molecular weights. Here lies a very impo~t point: if one wishes to "isolate" the topological effects in order to test molecular models, one has to use rheological functions defined at the same segmental mobility, and hence the same value of the mobility factor: as far as viscosity is concerned, the reduced function Tlo/M (T) will be used instead of the viscosity itself.

2.2.3 The effects of concentration (concentrated solutions): In the case of concentrated entangled solutions, the "elastic" parameters follow power law dependences as a function of polymer volume fraction r : 0 GN)sol=(GN)bulk

( 0

{~{~

( jO)sol=(jO)bulk ~ - a

(2-23) (2-24)

with an exponent a - 2-2.3 for entangled chains, so the product jo G~(which reflects the polydispersity of relaxation times) remains the same whatever the concentration. That means that the effects of the addition of small compatible species on the elastic parameters are mainly topological, i.e., the nature itself of the solvent molecules has a very small effect on the melt elasticity and the shape of the distribution of relaxation times. On the contrary, the effects of dilution on the polymer viscosity will be twofold : - a topological effect on the structure factor $ that will be described by molecular models; a change of the mobility factor M, that may either increase or decrease, depending on the plastifying -or antiplasfif3dng- effect of the molecules added to the polymer. -

2.2.4 The self-similarity of the viscoelastic behaviour of flexible chains The above phenomenological description of the viscoelastic behaviour of polymer melts and concentrated solutions leads to the following i m p o r t a n t conclusions 9if one focuses on the behaviour in the terminal region of relaxation, what is usually done for temperature (time-temperature equivalence) may also be done for the concentration effects and the effects of chain length; one may define a "time-chain length equivalence" and "time-concentration equivalence"[4]. For monodisperse species, the various shifts along the vertical (modulus) axis and horizontal (time or frequency axis) are contained in two reducing parameters: the plateau modulus G~ and a characteristic relaxation time, either Zw = qo jo or ZN = rio/G~. A plot of G*(cOZo)/G~ - where zo is either T~Vo r 1; s - in the terminal relaxation region is a universal function independent of temperature, concentration, chain length, and independent also of the chemical nature of the polymer (Figure 4). This self-similarity of the viscoelastic behaviour of monodisperse linear chains, whatever their chemical structure, may be extended to polydisperse species having the same shape of the molecular weight distribution (i.e., the same

105 polydispersity index as a first approximation). This implies some universality in the large-times relaxation processes of entangled polymers. As a consequence, the general features of the mechanical relaxation of long and flexible polymeric chains will be described by molecular models that do not "see" the local structure but describe the overall diffusion and relaxation of these chains in a universal way. Hence the power of the models described below lie in their universality:, it is easy to shift from one polymer to another, changing only a few parameters linked to the local scale structure of the polymer under study.

0-

o

@

-2

l

-I

I

0 log (~qo G~

I

I

,_,

I

2

Figure 4 : Master curve for the linear viscoelastic behaviour of entangled polymers in the terminal region of relaxation : V Polystyrene, bulk (M=860000, T=190~ Q Polyethylene, bulk (M=340000, T=130~ A Polybutadiene solution (M=350000, polymer=43%, T=20~ [from ref.4]. 3. THE CASE OF E N T A N G L E D M O N O D I S P E R S E L I N E A R S P E C I E S : PURE REPTATION

3.1. The basic reptation model The reptation concept was introduced by de Gennes [5] in 1971: it is based on the idea that long and flexible entangled chains rearrange their conformations by reptation, i. e., curvilinear diffusion along their own contour. De Gennes considered the reptation of a linear chain among the strands of a crosslinked network which create p e r m a n e n t topological obstacles. First, the dynamics of the wriggling motion of the chain along its own contour (what Doi and Edwards called later the primitive path) was described by de Gennes in terms of a diffusion equation of a "defect gas": he showed that this motion is fairly rapid: its longest relaxation time Teq is proportional to M 2, where M is the molecular weight of the chain (that time Teq would be equivalent to the ~B relaxation time in the slip-link model; see text below and Fig.8).

106

9

9

9

9

9

9

9

9

Q

9

Figure 5 9The basic concept of P.G. De Gennes 9reptation of a chain trapped in a tube-like region by migration of "defects" along the chain. At times t >Teq, the wriggling motion results merely in a fluctuation around the primitive path, so the chain moves coherently in a one-dimension diffusion process, k e e p i n g its arc length constant. The macroscopic diffusion coefficient of a reptating chain scales with chain length (molecular weight) as" D o, M-2,

(3-1)

and the time for complete rearrangement of conformation" 1; o~ M 3.

(3-2)

This time is the longest relaxation time of a linear chain. We will refer to it as the "reptation time".

e

,

~

-

,

.

e

/

.-

Figure 6 9The tube concept: The real chain is trapped between entanglements and is wriggling aroud the "primitive chain" (full line).

Doi and Edwards used this concept to derive the viscoelasticproperties of polymer liquids from the dynamics of reptating chains [6]. They ass1~med that

107 reptation would be the dominant relaxation process for polymer melts, even in the absence of a permanent network. This is a very strong assumption, as the topological constraints are made by surrounding chains that also diffuse by reptation. This assumption was justified later when analyzing the constraints release (or tube renewal) process (section 4).

~/

I "- \ ?

J

f /

~\

\ __kJ

Figure 7 9Reptation: the chain disengages from its initial tube by back-and-forth motions; the time necessary for a complete renewal of its initial configuration is the "reptation time" which is the longest relaxation time of the polymer. When analyzing the overall diffusion of a single chain due to Brownian motion, the topological constraints made by the surrounding chains confine the chain in a tube-like region. The centreline of the tube is called the "primitive path" and can be regarded as the curve which has the same topology as the real chain relative to the other polymer molecules; the real chain is then wriggling around the primitive path. The Doi-Edwards calculation is based on the theory of rubber elasticity. In order to calculate the time-dependent properties, the contribution of individual chains to the stress following a step strain is evaluated; then the relaxation of the stress is related to the conformational rearrangement of the chains by a reptation process. For this calculation, the topological constraints along the chain are represented by frictionless rings around the chain (Fig. 8), which is another way of describing entanglements. The succession of segments ("primitive segments") joining these "slip-links" along the chain is called the "primitive chain". As the sliplinks are now local constraints, the continuous nature of the constraints is accounted for by assuming a natural curvilinear monomer density along the chain. So all three models (reptation among fixed obstacles/tube model/slip-links model : Figs. 5 to 8) are equivalent in essence. When submitting the polymer sample to a sudden deformation (step strain), the primitive path is distorted by affine deformation (Fig. 8b), and the curvilinear monomer density is perturbed from its equilibrium value.

108

B

D

A

C

a) t < O : e q u i l i b r i u m

A

C

F

E

c) t--_ 1;a ( A p r o c e s s )

state

E

b) t = O : s t e p s t r a i n C

E

d) t _=_ ~s (B p r o c e s s )

C~

B'

E~

D'

e) t =_ 1:c (C p r o c e s s : reptation)

Figure 8 " Relaxation of a polymeric chain after a step-strain deformation[6]: process A (8c): reequilibration of chain segments; process B (8d): reequilibration across slip links; process C (8e): reptation. Then the chain will relax: The f i ~ t relaxation process (called the A relaxation process; Fig. 8c) which occurs at the shortest times will be a local reequilibration of monomers without slippage through the slip-links. In other words, it is basically a Rouse relaxation process between entanglement points which are assumed to be fixed in that time scale. The characteristic relaxation time of this process is rather short and is independent of the overall chain length (see below). The second relaxation process (B process; Fig. 8d) is a reequilibration of segments along the overall chain, i.e., across slip-links. It is basically a retraction of the chain to recover its natural curvilinear monomer density, which may be depicted as a Rouse relaxation process along the entire chain. In the last relaxation process (C process; Figs. 7 and 8e), the chain renews its entire configuration by reptation. The viscosity, p l a t e a u modulus, limiting compliance and m a x i m u m (terminal) relaxation time derived from the basic D-E model are power laws of the molecular weight M: Tlo or M 3, G~ or M o, jo or M 0, to = ~o jo o~ M 3,

(3-3)

109

and

jOG~=6. 5

All viscoelastic functions may be expressed in terms of a single reptation parameter (for example the plateau modulus or tube diameter) and the monomeric friction coefficient (or mobility factor in our terminology), in agreement with the above phenomenological presentation. If the general features of the viscoelastic behaviour are well depicted, the experimental molecular weight dependence of these parameters is : 110or M3.4, G~ o~ M o, jo o~ M o,

(3-4)

~0 = T10jo or M3.4, and j O G ~ = 2 . In particular, the fact that the experimental "polydispersity of relaxation times" jo G~ is larger than the theoretical value indicates the presence of other relaxation processes. In the following section we will describe a somewhat different analytical derivation of the Doi-Edwards model taking into account additional relaxation processes (in particular the Ta (glass transition) relaxation). Our derivation, which gives a good quantitative agreement with the observed linear viscoelastic behaviour for a large number of linear polymers, keeps however the basic physical concepts of the reptation model. Hence this model is dedicated only to the case of entangled linear monodisperse species. The case of branched polymers, polydispersity and short chains effects will be presented in, respectively, sections 4,5 and 6. The same basic models are also used in chapter 1.1 and 1.3. In chapter 1.1, the mutual diffusion in polymer melts is related to the reptation and constraint release processes, whereas in chapter 1.3 the relaxation of chain segments along the polymer chain is investigated in terms of the some relaxation mechanisms as described below. 3.2. Detailed d e r i v a t i o n of linear viscoelastic p r o p e r t i e s of l i n e a r s p e c i e s [7] 3.2.1. The (A) relaxation process That relaxation process may be defined as a Rouse diffusion between entanglement points. The characteristic relaxation time of the (A) process is : ~ob2 N 2, 1;A = 6~2ksT

(3-5)

where Ne is the number of monomers between e n t a n g l e m e n t points, ks Boltzmann's constant, T the absolute temperature, b the effective monomer length

110 (b=C~o 1,1 being the monomer length) and ~o the monomeric friction coefficient. The relaxation function associated to the (A) process is :

FA(t) =

expF-L P~At],J

(3-6)

p=l

where Ne is the number of Kuhn segments between entanglements. In order to define completely the relaxation function, we have to determine the initial modulus

G'N SO: O(t) = GN FA(t),

(3-7)

G'N is a function of strain and may be written as : GN = pRT

1

Me (~Ul>o ,

(3-8)

w h e r e p is the polymer density, E is the strain tensor, u a unit vector corresponding to the vector linking two entanglement points (slip-link segment) [6]; 1 the < ~ / o term expresses the fact that the molecule is outside its equilibrium configuration" from Doi-Edwards theory G N = 4 pRT.

5 Me

We may then write the relaxation modulus corresponding to the (A) relaxation process (short times) as :

N~e x p [ _ p t21 GA(t) = 4 pRT ~ 5 M e p=l L ZA J"

(3-9)

3.2.2. The (B) relaxation process Doi and Edwards postulate that the chain recovers its equilibrium monomer density along its contour by a retraction motion of the chain within the tube. That motion, induced by the chain ends, leads to a relaxation function : 8

FB(t) =

poddZp2/t;2

exp[-p2t] L -~-BJ'

(3-10)

with: ~0 b2N2 ~B =

3g2kBT

where N is the number of segments along the chain.

(3-11)

111 Viovy [8] describes that re-equilibration process as an exchange of monomers between neighbouring segments : he calls that process "reequilibration across sliplinks", and the corresponding relaxation function may be written as : N/Ne Ne [ p2t] FB(t)= ~ - - ~ - e x p - - - - - . p=l

(3-12)

~;B J

This function corresponds to a Rouse spacing of relaxation times and gives a better fit of the experimental data than Eq. 3-10. Hence the relaxation modulus of the (B) process may be written as a hmction of the entanglement density N/Ne:

G B(t)= 4 pRT N~~ Ne

5 Me

[ p2t]

-'NeXPL-Bj

(3-13)

3.2.3. Reptation: relaxation process C In that final relaxation process the molecule recovers its final isotropic configuration by a reptation motion. The characteristic time for the reptation process is (see also chapters L1 and L3):

1 ~0a2Ne(N) 3 ~:c =--g kBT "~e '

(3-14)

where a is the tube diameter (a 2 = Ne b2). The relaxation function is given in the original Doi-Edwards picture [5-8] by the equation" Fc(t)= E p28~2exp[-p2t]

podd

(3-15)

k "~'CJ

The plateau modulus of that relaxation domain is :

=RT

(3-16)

Me ' and hence the relaxation modulus is: Go(t)= pRT

8 exp[-pat] podd

(3-17)

k -~-cJ

The refinement introduced by Doi [9] who considers tube length fluctuations is more relevant to experimental scaling laws as far as the viscosity/molecular weight dependence is concerned. Following that concept, Gc(t) can be cast into the integral form :

112

Oct':0RT[ Me JO4N/N~

e

'/ '

/

- ~ ( i i d~ + ~2v/s, exp - ~(2i

~,'

(3-18)

where N ~4v4 ~(1) = Ne 16 Zc

2v ~ < ~]N/Ne,

v

~(2) =

~ - 4N / Se

1 > ~ > ~~/'N/N~, 2v

~c

(3-19)

(3-20)

The v parameter may be approximated to 1 for highly entangled chains.

3.2.4. The Ta high-frequency relaxation domain: the transition region between the rubbery and glassy regions In order to give an analytical representation of the mechanical properties in the high frequency range characterizing local motions in the molecular chain, we used analytical forms derived from studies on dielectric relaxation : a Cole-Cole or Davidson-Cole equation generally gives a good fit in the transition region; we used in the present case a Davidson-Cole equation, that presents the advantages of being truncated at large times and to give analytical forms both in the time and frequency domains" G *HF = Goo -

Go. (1+ j ~ H F ) 1/2'

(3-21)

with"

1;HF -

~o 12 X2kBT

(3-22)

The inverse Fourier transform of Equation 3-21 gives the relaxation modulus"

]

(3-23)

Other m a t h e m a t i c a l forms may be used to describe the high frequency relaxation. These various equations, either phenomenological or based on diffusion defect models lead to a characteristic relaxation time ~;HFof the glass transition (Ta) domain of the same order of magnitude.

113 As a s u m m a r y , the characteristic relaxation times of the various relaxation mechanisms presented here above are linked to each other by (see also Chapter

1.3):

I:.~ = ~1 I:i Ne2 SB = 21;A( ~ ) 2 = ~l 1;i N2 N

(3-24)

N3

1;c = 3ZB ~ee = I;i Ne with ~i = ~0 b2 ~2kB T " 3.2.5. Viscoelastic function in the whole time/frequency domain Thus the relaxation modulus may be calculated from a very limited number of physical parameters (G ~ Goo and ~i), with no "ad-hoc" parameters, in a time range covering the initial glassy behaviour down to the terminal relaxation region. For a typical polymer, this range exceeds ten decades of times. The complete expression of the relaxation modulus is :

NINe exp - zr

G(t) = Me

+-

2v e exp 4N/N

~ 2)

[ p2tll + G~ [ 1 - e f t [ p2t 1 N~eNe exp--+ --~-exp5 Me L p=l I:AJ p=l ~BJJ

4pRTIN~e

(#--~) ] .

(3-25)

The complex shear modulus is the Fourier transform of Equation 2-25 : G * (co) = G~

2,,

f04N/Ne

jcozr 1+ jco~(1)

d~ +

~1

jco~r ] 2v d~ ~]N/ Ue 1+ jC0Z~(2)

NNe

+ -G~ e Ne jc0(1:B/p2) + jc0(~A/p2) 5 k p=l N l+jco(~ B/p2) p=l 1+ jco(xA/p2)

] I

+ Goo 1-

1 1

1 .(3-26) (1+ jO~HF)2

We have reported on Figs 9 through 12 a comparison between the experimental complex shear modulus and its theoretical calculation (full line) for two polymer species. The model fits reasonably well the linear viscoelastic properties of a large number of linear polymers ranging from polyolefins to glass-forming polymers. This calculation gives us the basic "long-chains monodisperse behaviour" which feeds the more complete derivation taking into account the effects of constraints release (section 5).

114

,oj.

1

f

8.-

- 8I

Eo

~-6

~,6 = o

4

4

i

-5

0

l

log t~ (sec -I]

5

I0

log ~

(sec-al

Figures 9 and 10 9storage (G') and loss (G") moduli of nearly monodisperse polystyrene samples at 25~ 9(A)M=900000; (Q)M=400000; (O)M=200000; (0) M=90000; full line ( )" theory (eq.3-25) [from ref. 7].

I~ f

/ 8~-

:

~'

I

1

i

"

'l

1

1

Io

l

~

;

~

~

;

J

!i 1

8

4 i

loq ~

(sec J]

log ~J (sec -~)

Figures 11 and 12: Storage (G') and loss (G") moduli of nearly monodisperse polybutadiene samples at 160~ (A)M=361000; (~)M=130000; (O)M=39400; full line ( ): theory (eq.3-25) [from ref.7]. 4. E N T A N G L E D M O D E L . B R A N C H E D POLYMERS Branched polymers may be classified into two categories from the point of view of rheology : - polymers with short branches (Marm"dt

=

(5-13)

~Z[P(M)exp(-~/z~r

which yields the following tube renewal time for each N-chain in the sample" zt(M, P(M)) =

(5-14)

< Zcr >

The expression of < ~c, > has been checked for binary blends of monodisperse polystyrenes (Fig. 21). The tube renewal time of the high N-component is measured at different volume fractions CN and the molecular weight distribution is defined by two step functions : 1-CN at Mp and r at MN. The experimental data fit well the model with z=3.

f0

_

//

--

//

~

,~ i[ ."/( /'.," / II I

0

0.5 q>N

Figure 21: Variations of the tube renewal time of N-chains (MN = 2 700 000 g.mo1-1) in a matrix of shorter chains (Mp = 100 000 g.mo1-1) of polystyrene, as a function of concentration ~N [21].

126 Therefore, the longest relaxation time of a chain in a polydisperse somple is modified by a shift factor ~. defined by:

~'(M) =

'r(M, P(M)) 'rtl(M:) + zc(M) -1 z(M) = -~ = 'rren(M'P(M))+'rc(M)-I

3/-----~ 4M §1 2 .... 9 're ( - - ~ ) + 1 is lower than Zcr(M), the relaxation time is decreased (~.< 1) whereas for shorter chains the relaxation time is higher in the blend than in a monodisperse environment. 5.2.2.2. Relaxation functions The double reptation approach allows us to visualize the blend of n different species of the s~me polymer (molecular weights : M1, M2, ..., Mi, ... Mn ; volume fractions r as a network of (i, j) knots accounting for the entanglements between an i-chain and a j-chain [14, 19, 22, 27]. Therefore, the time-dependent density Fij (t) of initial knots in the blend is proportional to the relaxation function of each species involved in the knot, t h a t is to say : Fi5(t) a Fi(t) Fj(t).

(5-16)

From t h a t relation, it can be shown t h a t the density of (i, j) knots is equal to the geometric average of the density of knots between similar chains : (5-17)

Fij(t) = [Fii (t).Fjj(t)] 1/2

As the volume fraction of (i, i) knots is ~i2, and that of (i, j) knots is given by 2 {~i.{~j, the overall average number density of initial knots can be written as : ~

F(t) = Z Z r

(t)= [Z r

1/2

2

(t)].

(5-18)

On the other hand, the relaxation function of (i~i) knots is directly connected to that of i-chains by a mere shift, e.g. : Fii (t) = F i (at). In other words, in a monodisperse polymer the knots are renewed at a rate proportional to t h a t of the chain segments. The shift coefficient a is assumed to be length-independent, allowing the same shift factor to be applied to the overall relaxation function F(t) = F(at). Then, we recast relation (5-18) into :

F(t)=

r

"

(t)

.

(5-19)

127

The individual relaxation function Fi (t) is defined from Doi's expression (relation 3-18) where Fi (t) = Gc(t) / G~ and % is replaced by: x(M,,) = [zj1 + (Me/M)2(%r)-l] -1 accounting for the molecular weight distribution. An experimental check of such a quadratic blending law is given by the storage modulus G'(m) of binary blends which exhibits a plateau G'N at intermediate -I -i, frequencies, 2;(M1) < (DO < 1~(M2) corresponding t o $(M2) < 1;0 < 2;(M1) for the relaxation function. Therefore, for blends such as M 1 >> M~, the blend relaxation function is given by F(to)_--r 2 leading to G'N = ~12 G~, which is observed experimentally [28]. For a polydisperse polymer defined by its MWD function P(M), the relaxation function is given by"

F(t) =

P(M)

2(t,M)d In M

(5-20)

if only the reptation process is taken into account. But, for large polydispersities, the Rouse process (B) of the long chains overlaps the reptation process of the short chains. Consequently, the most general expression of the relaxation function (or relaxation modulus) must include all the relaxation processes described in part 3.2. As the Rouse dynamics is assumed to be linear with respect to the MWD and that the A and HF processes are mass independent, we define the relaxation modulus of a polydisperse linear polymer by :

[f

G(t)= +~P(M) G~I2 (t,M)dlnM

+

(5-21)

P(M)GB(t,M)dlnM+GA(t)+G~(t),

which is consistent with rel. (3-24) for monodisperse samples. 5.2.2.3. Viscoelastic behaviour The relaxation modulus is the core of most of the viscoelastic descriptions and the above expression can be checked from experimental viscoelastic functions such as the complex shear modulus G*((D) for instance. In addition to the molecular weight distribution function P(M), one has to know a few additional parameters related to the chemical species :the monomeric relaxation time xo, the rubbery plateau modulus G~ and the glassy plateau modulus G~. The temperature dependence is included in the relaxation time Xo and more precisely in the friction coefficient ~o- Expressed in terms of free volume fraction f which increases linearly with temperature and expansion coefficient af, the WLF equation gives the temperature dependence from two parameters C1 and C2 o

o

128 at a reference temperature To. The product C~ C2 = B / af is constant as long as the free volume expansion factor af can be considered as temperature and mass independent. An alternative description such as the Vogel equation introduces a temperature T~ = TO- C 2 which is a constant for a given high polymer species. Therefore, the friction coefficient can be written as : ln~o = lnA +

B

af(T - T.)

.

(5-22)

The values of B / o~f and To. are tabulated for different polymers [29] and the value of A can be derived from the elementary relaxation time zi measured in the transition zone. The high-frequency domain does not depend on molecular weight value and distribution, and thus the tabulated values of ~o at a given temperature are applicable to commercial samples. Figure 22 gives two examples of the description of the viscoelastic data of commercial polypropylene and high-density polyethylene samples by the expression for the complex shear modulus derived from expression (5-21).The first term is dominant for highly entangled systems.

~3

2_

I i

+.§

IoQw

Figure 22 : Experimental data and theoretical curves (expression 4-21) of the complex shear modulus of commercial polypropylene (M w = 348 500, Mw/M N = 6.1) and high density polyethylene (Mw = 210 000, Mw/M s = 11.7) [19] The agreement is satisfactory but it is worth noting that the fit will be poorer if the high molecular tail is not described properly or more generally if the relaxation time shi~ function ~(M) is not correct. For example, we showed [19] that failure to take into account the shift; factor k leads to a large discrepancy between the model and the experimental data.

129 Another important point is that, when approaching Me, the tube consistency becomes weaker or in other words, the constraint release scaling law is modified and the rubbery plateau disappears whereas the steady-state compliance jo decreases. A self-consistent approach should predict that around Me, the reptation modes would be gradually replaced by Rouse modes in order to describe the non entangled- entangled transition. 6. E F F E C T S OF NON-ENTANGLED CHAINS

6.1. The unentangled r e g i m e It is commonly admitted that a linear flexible polymer melt behaves as a dilute solution as long as the molecular weight is sufficiently low so that entanglement effects do not occur. The Rouse formulation of the bead-spring model with no hydrodynamic interactions holds for such undiluted polymers because of the presence of segments belonging to other chains within the coil of a given molecule. The Rouse description predicts a relaxation modulus given by GB(t), Equation (313), where the product MeN/Ne is replaced by the molecular weight M, so the longest relaxation time is : ~oR2N

6M

(6-1)

"~Rouse - 6~2kT - 11o p~2NART,

Rg being the radius of gyration and NA the Avogadro's number; it follows that : ~oR2N A 0 0.4M. no = P 3-6~oo and Je = pRT ~

(6-2)

The temperature dependence of Tlo is mainly included in the friction coefficient ~o (relation 5-22). Therefore, 11o can be expressed by : In 11o= ln($(M)) +

B o~f[ T - Too(M ) ]

,

(6-3)

where the structure factor $(M) describes the variations of the radius of gyration. Furthermore, the temperature Too is no longer a constant. In the free volume models, T~ accounts for the variations of the free volume fraction f (f = a f ( T - Too)) which is assumed to be mainly due to the concentration of chain ends. As the chains become shorter, the free volume fraction f increases, hence Too decreases.

130

2O

I

1

i

I

$

15_~I0

I

-

I/) 0 (J

._. > 5-

+ 0 0

1

I

50

I

I

1

I00 150 2 0 0 2 5 0 3 0 0 temperoture (~

F i g u r e 23 9D a t a of zero-shear viscosities of polystyrene fractions ranging from 900 g.mol -z to 30 000 g.mo1-1 as a function of t e m p e r a t u r e [29-37]. The m a s t e r curve is obtained by experimental shifts from the data of a reference mass of 110 000 g. mole -z. It includes more t h a n one hundred experiments lying within the experimental b a r error. A least squares analysis gives the p a r a m e t e r s of the reference mass and the other ones are deduced from the shiit factors. The plot of a m a s t e r curve of the thermal variations of 11o for various molecular weights and temperatures (Fig. 23) shows that the expansion coefficient af can be considered as a constant in a wide range of t e m p e r a t u r e s . The vertical and horizontal shift factors respectively describe the mass dependence of the radius of gyration and temperature T.. Polystyrene is a good example for analyzing the non-entangled regime because the molecular weights available are as low as Me/20. Consequently, the experimental data are significant in a range of molecular weights exceeding one decade.

/ v

,

,

,

v

4O

z:: 01--I0~

0

No I

0.2

I

0.4

! "~

0.6

103/Mw

I

0.8

,

J

i. i.0

F i g u r e 24" The horizontal shift factors of the master curve of Fig. 23 give the t e m p e r a t u r e Too as a function of molecular weight (reference Too = 49.4 ~ C).

131 The variations of T=. are derived from the horizontal shift factor and can be expressed by (Fig. 24) : D T.. = (T..)= - - M

(6-4)

For polystyrene, D = 83 500g.mol-t, which means t h a t beyond a mass of approximately D, the temperature T=. is fixed - (T=)=. = 49.7 _+0.3~ in the entangled region, the free volume fraction is constant at a given temperature and the iso-free volume state merges into the isothermal state. From the vertical shift factor of the master curve, we are able to describe the mass dependence of the zero-shear viscosity in the iso-free volume state which is directly connected to the radius of gyration of the chains. In the molten state, it is generally assumed that the chains exhibit a Gaussian conformation and therefore the viscosity should be proportional to the molecular weight. Unexpectedly, we observed (Figure 25) an unambigous different scaling law (% a M 6/5) which confirms previous results [31] and should be explained by the variations of the radius of gyration. This hypothesis is consistent with direct measurements of Rg by SAXS experiments on solutions of polystyrene in O conditions [38]; the same scaling law is found for molecular weights lower than about Me (Fig. 26). 104 _

I0 ~ ~r

Io' . "" 4 t

6/5

sO

,o~

i/

I01 103

I

104 Mw

IO s

Figure 25 9the vertical shift factor of the master curve of Figure 23 gives the structure factor A' as a function of molecular weight. In addition, Pearson [39] made numerical simulations of the mean-square radius of gyration of polyethylene by using a rotational isomeric state method. For nalkanes and low molecular weight polyethylenes below M c, he also found a stronger increase of the calculated radius of gyration with molecular weight than expected from Gaussian statistics. Therefore, Gaussian statistics does not seem to apply to short chains as shown by numerical simulations [40] but the observed

132 scaling law, which is the same as for solutions in a good solvent has no connection with an expansion of the chains due to excbaded volume effects as the absolute value of Rg is lower than the extrapolated gaussian value (Fig. 26).

A

..,~ IOs r

I

i

10 4 -

_

a./-

/a

.y

N~

/O

-61'

I0 2/

/ ~l/._j 615 I~ E!

I 10 4

10 3

I

IO s

Nw

Figure 26 : Molecular weight dependence of the mean-square radius of gyration of solutions of PS in cyclohexane at 34.5 ~ C [38]. The steady-state compliance Je~ follows the Rouse expression until M'c --5 M e. Then, the longest relaxation time is expressed by : 15 Tlo j0 ZRouse -

~2

(6-5) "

From dynamic experiments and applying the time temperature superposition principle, the complex shear modulus is measured over about five decades and the Rouse model can be checked extensively [37]. I0 ~

I0 e ,,,,..,

iO s na 10 4

b I0 2 IOOi I0 ~

/i

1 I0 2

I

i I0 4

1 c~(s -4)

1 I0 e

I

1 I0 e

~

J I0 I~

Figure 27 9Experimental complex shear modulus of unentangled polystyrene (M = 8 500 g.mol-1) compared to the Rouse model [37].

133 The agreement is good over the whole range of molecular weights below M e (Figure

27) provided that one adds the high frequency term (equation 3-21) accounting for the very local relaxation modes of the chains. The overlap between the two domains becomes significant at frequencies corresponding to 1 000 zi and 100 ziaccording to the molecular weight because of the very different orders of magnitude of the rubbery and glassy plateau moduli.

6.2. Effect o f s h o r t c h a i n s i n t h e e n t a n g l e d r e g i m e When short M S chains are introduced into a sample of entangled M L - chains with a volume fraction r of long chains such as r M L > M e, the blend can be viewed as a concentrated solution of the long chains, or in other words, the M s component is acting as a solvent at least in the terminal zone of relaxation of the long chains. According to Doi-Edwards theory, the reptation of the long chains will occur in a tube whose diameter a varies as r Thus the number of m o n o m e r s between entanglements will scale as r Accordingly, the reptation time zc (relation 3-14) should be proportional to r as a first approximation, the zero-shear viscosity Tlo and the steady-state compliance j0 should respectively scale as r and r E x p e r i m e n t s conducted on concentrated solutions of high molecular weight hydrogenated polybutadiene in a commercial oil [41] showed t h a t the expected law for 11o and the average terminal relaxation time corn1 is satisfied (Fig. 28).

107

I

"1

I '

I0 e 0

I0 ~

!

I'

I

,E

10 5

iO-I

10 4 10 3 _l,~m IO'J

[

j

j (JD

!

l I

lnLn

I0 "j

I

n

n

n

~

t

J

I

Figure 28 9Zero-shear viscosity and average relaxation time of concentrated solutions of entangled hydrogenated polybutadiene (M/M c = 300) [from ref 28]. The e x p e r i m e n t a l scaling laws are T10 a r and corn1 a r in a g r e e m e n t w i t h DoiEdwards' theory.

134 For melts of long chains containing short chains, the contribution of the unentangled component is in some cases non-negligible and has to be removed from the data of the blend in order to isolate the contribution of the entangled component:

TIoL = ~lO - (1- r

(6-6)

and JeOL= J ~ TI_x_O)2, 1]OL

(6-7)

as defined for concentrated solutions. Moreover, as far as relaxation times are concerned, the reptation mechanism in an enlarged tube should lead us to favour the tube renewal process. The expression of the tube renewal time Zren (relation 5-8) shows a ~3 scaling which implies that ~ren(ML,~)=zc(ML,~) for M=(Me/2dp) 4/3. Therefore, the double reptation approach applies in an extended range of molecular weights because the concentration is low. The overall relaxationtime z(M L, ~) can be cast into the form

,~(ML ' ~)-1 = [lii ,I;c (M)]-I + [{il3Zren (M)] -1

or 1;(ML,~))= ~ zc(ML) [1+ 4~2ML 3/2M2 ]-1,

(6-8)

whereas the zero-shear viscosity TIoLwill vary as r z (ML, 4)). The experimental evidence of the importance of the tube renewal mechanism when short molecules are added to a high polymer is provided by blends of narrow polystyrene (M s = 8 500 and M L = 900 000) [28]. In a large range of concentrations (4)> 0.05) where the high component is assumed to behave as an entangled melt, the variations of the terminal relaxation time ~m-: in the iso-free volume state (Fig. 29) confirms relation (6-8). As the steady-state compliance J~L scales as ~-2 (Fig. 30), the zero-shear viscosity T10L varies as expected and the plateau modulus G~ which reveals the entanglement network is proportional to r The description of the relaxation modes including the behaviour of the entangled M L - chains and unentangled M S - chains remains to be done. At high concentrations, the long chains will dominate the viscoelastic behaviour in the terminal zone but, when approaching 4) M L = Me, the tridimensional diffusion of the short chains impeded by the surrounding long molecules will bring a noticeable contribution which could depart from the Rouse description.

135

10 3 s s

._E i0 z

==-

s

!

I0 i I0 o .i.,,

10-2

, i

~i

l.,lll

i i

I

I0 ~

I

I

@

Figure 29 : Average relaxation time of a high molecular weight polystyrene (M = 900 000) in the presence of short chains (M = 8 500). The dotted line represents pure reptation and the full line stands for the contribution of tube renewal according to relation (6-8).[from ref. 28]

10-3

I

I

10-4 -3

lO-S

I0"6

lit,,

t

IO"~

~

i

i

i

l

I

Figure 30 9Steady state compliance as a function of concentration (same blends as in Fig. 29). 7. P R O B L E M S STILL P E N D I N G Along the same lines as described above, several questions arise as far as molecular weight distribution is concerned. The main question is : in the near future, will we be able to predict the viscoelastic behavior of linear polymers w h a t e v e r the molecular weight distribution, encompassing the broadest dispersion from oligomers to very long molecules ? The

136 answer could be positive provided t h a t we know more about the environmental impact on the relaxation modes of each individual chain, whatever its molecular weight. Remembering t h a t three characteristic molecular weights are defined from viscoelastic parameters : Me, the molecular weight between entanglements, M c _=2 - 3 Me, the cross-over mass for viscosity and M'c = 5 - 7 M e, the cross-over mass for compliance, intermediate situations should be explored which could be, in the last stage, merged into a "universal" law : All the masses lie beyond M'c : t h a t situation has been described in this chapter and it could easily be extended down to Mc by taking into account the decrease of the compliance, All the masses lie below M e : in the literature, the Rouse description of the relaxation modes of non-entangled melts or solutions is also used for polydisperse samples by means of a linear blending law. In order to consider the free volume variations of each mass in the blend, the relaxation times have to be shii~ed by a factor ~. which is the ratio of the monomeric friction coefficients of the blend ~Ob and of each species (~o)For a polydisperse sample [42], the relaxation modulus can be cast into : G(t) = I Z P ( M ) G ( k t , M ) d l n M ,

with

(7-1)

k = ~o ~Ob

As a consequence, the zero-shear parameters are : Rg2(M) d i n M tlo b = P 36Na m o ~0 b l'~p(M) -~

(7-2)

0.4 and jo = pRT rl--'-----~K M

(7-3)

P(M)R 2 (M)d In M.

Therefore, for Gaussian molecules, the above parameters are functions of moments of the molecular weight distribution 9tl0 a M w and j0 a Mz.Mz+l /Mw. Otherwise, the mass dependence should be slightly different for tl0 and a large deviation from a combination of various average molecular weights is expected for the steady-state compliance. For mass distributions extending over the cross-over region between the nonentangled and entangled regimes, the situation is more complicated as anticipated in section 6. When the average molecular weight Mw - or other m o m e n t of the distribution - is lower than Me, a Rouse diffusion could be expected with relaxation times shifted according to the free volume variations, even for the high part of the distribution. Reciprocally, when the average molecular weight is higher than M c,

137 the reptative motion of the long chains will be made easier by the short chains according to the description given in section 6. For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeaux [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time ~i, proportional to M 2, which can be considered as the Rouse time of an entangled polymer moving in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced ~i by an empirical combination of 1;i , ~c a n d

1;Rous e .

In a more empirical way, Lin [43] establishes an expression of the stress relaxation modulus of monodisperse polymers including the Rouse motions between entanglements and a relaxation process related to the slippage of the polymer chains through entanglement links. The associated relaxation function is a single exponential with an empirical relaxation time ~x- Moreover, the reptation process is assumed to be accompanied by a fluctuation one relevant to a Rouse description and which is the only mechanism remaining at M = Me. Consequently, he qualitatively describes the entire range of molecular weights including the entangled and unentangled regimes. Therefore, more work remains to be done in order to interpret the viscoelastic behaviour of very broad polymers with average molecular weights ranging from low values, lower t h a n Me, to strongly entangled situations. A comprehensive MWD - viscoelastic properties relationship will be of great help for designing materials for specific applications but a new challenge has been around for the last few years. It consists in developing numerical and analytical methods to invert a linear viscoelastic material function to determine the molecular weight distribution. There are several reasons to pursue such an objective - m a n y commercial polymers are slightly or not at all soluble in usual solvents, thus techniques like gel permeation chromatography or light scattering are inapplicable. Rheological characterization can be performed on-line and in real time and it is also a less cots-effective technique. So far, the important issue on how to determine MWDs from rheological data has been addressed with limited success, mainly for three reasons. The monodisperse relaxation function F(M,t) must be described precisely in the terminal and plateau regions, one has to provide a correct blending law yielding the polydisperse relaxation modulus G(t) ; and even if these two obstacles are overcome, specific mathematical procedures are needed in order to solve the illposed problem of numerical inversion of integrals. Many different sets of solution parameters are not physically m e a n i n g ~ l and appropriate constraints have to be imposed in order to determine an acceptable MWD. For entangled systems, the two first conditions are fulfilled in the framework of reptation theories : a comprehensive expression of the monodisperse relaxation modulus G(M,t) is given by expression 3-24 and the double reptation model generalized to a continous molecular weight distribution provides the integral relation between the MWD function P(M) and the polydisperse experimental

138 complex shear modulus G*(co) derived from the polydisperse relaxation modulus G(t) (equation 5-21). A recent publication by D.W. Mead [44] investigates numerical and analytical methods involving the double reptation mixing rule used with specific relaxation functions. A single exponential or a step-function monodisperse relaxation function are relevant to analytical methods whereas more general multiple time constant monodisperse relaxation functions require numerical treatments. The first results sound encouraging, making t h a t rheological measurements which imply bulk, macroscopic techniques are able to catch important molecular features such as chain length distribution. This is further evidence of the sensitivity of rheological techniques to nanoscopic changes, moving them into analytical tools.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

B. Gross "Mathematical structure of the theories of viscoelasticity", Hermann. Paris (1953). G. Marin, J.J. Labaig, Ph. Monge. Polymer, 16 (1975) 223. J.P. Montfort, Doctoral Thesis, Universitd de Pau (1984) G. Marin, J.P. Montfort and Ph. Monge. Rheol. Acta, 21 (1982) 449. P.G. de Gennes, J. Chem. Phys., 55 (1971) 572; see also "Scaling concepts in Polymer Physics", cornell U. Press, Ithaca (1979). M. Doi and S.F. Edwards ' ~ e theory of polymer dynamics" Oxford U. Press, Oxford, (1986). A. BenaUal, G. Matin, J.P. Montfort and C. Derail. Macromolecules, 26 (1993) 7229. J.L. Viovy J. Polym. Sci, Physics, 23 (1985) 2423. M. Doi J. Polym. Sci. Lett., 19 (1981) 265. T.C.B. Mac Leish, Europhys. Lett., 6 (1988) 511. P.G. de Gennes, J. Phys. (Paris), 36 (1975) 1199. D.S. Pearson and E. Helfand, Macromolecules ,17 (1984) 888. R.C. Ball and T.C.B. Mac Leish, Macromolecules, 22 (1989) 1911 J. des Cloiseaux, Europhys. Lett., 5 (1988) 437. J. Klein, Macromolecules, 11, (1978) 852, ASC Polymer Prep., 22 (1981) 105. M. Daoud, P.G. de Gennes, J. Polym. Sci., Phys-Ed., 17 (1979) 1971. W.W. Graessley, Adv. Polym. Sci, 47 (1982) 67. H. Watanabe, M. TirreU, Macromolecules, 22 (1989) 927. P. Cassagnau, J.P. Montfort, G. Marin, Ph. Monge, Rheologica Acta, 32 (1993) 156. H. Watanabe, T. Sakomoto, T. Kotaka, Macromolecules, 18 (1985) 1436. J.P. Montfort, G. Marin, Ph. Monge, Macromolecules, 17 (1984) 1551. P. Cassagnau, Doctoral Thesis, Pau (1988). J. Roovers, Macromolecules, 20 (1987) 184. J. Klein, Macromolecules, 19 (1986) 1. G. Marin, J.P. Montfort, Ph. Monge, J - N. Newt. Fluids Mechan., 23 (1987) 215. J. des Cloiseaux, Macromolecules, 25 (1992) 835. C. Tsenoglou, Macromolecules, 24 (1991) 176. J.P. Montfort, G. Marin, Ph. Monge, Macromolecules, 19 (1986) 393 and 1979.

139 29 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44.

J.D. Ferry Viscoelastic properties of polymers, Wiley, New York 3rd ed. (1980). G.C. Berry, T.G. Fox, Adv. Polym. Sci., 5, (1968) 261. R. Suzuki, Doctoral Thesis, Strasbourg (1970) unpublished results. S. Onogi, T. Masuda, K. Kitagawa, Macromolecules, 3 (1970) 1098. D.J. Plazek, V.M. O'Rourke, J. Polym. Sci., A9 (1971) 209. R.W. Gray, G. Harrisson, J. Lamb, Proc. Roy. Soc., London, A 356 (1977) 77. T.G. Fox, V.R. Allen, J. of Chem. Phys., 41 (1964) 344. J.P. Montfort, Doctoral Thesis, Pau (1984) unpublished results. J.C. Majestd, A. Benallal, J.P. Montfort, submitted to Macromolecules. T. Konishi, T. Yoshizaki, T. Santo, Y. Einaga, H. Ysmakawa, Macromolecules, 23 (1990) 290. D.S. Pearson, G. Ver Strate, E. Von Meerwall, F.C. Schilling, Macromolecules, 20 (1987) 1133. C. Degoulet, J.P. Busnel, J.F. Tassin, Macromolecules, 35 (1994) 1957. V.R. Raju, E.V. Menezes, G. Matin, W.W. Graessley, L.J. Fetters, Macromolecules, 14 (1981) 1668. J.C. Majestd, J.P. Montfort, submitted to Macromolecules Y.H. Lin, Macromolecules, 17 (1984) 2846. D.W. Mead, J. of Rheology, 38, 6 (1994) 1797.



141

Experimental validation of non linear network models

Christian CARROT, Jacques GUILLET, Pascale REVENU, Alain ARSAC Laboratoire de Rhdologie des Mati~res Plastiques. Universitd Jean Monnet. Facultd des Sciences et Techniques. 23 Rue Dr Paul Michelon. 42023- Saint-Etienne. Cedex 2. FRANCE

1.INTRODUCTION 1.1.Constitutive and model equations for computer simulation of complex flows. Nowadays, industrial and research applications of computers are widely used in the field of polymeric materials. Computer aided rheology and simulation of processing operations involving the flow of polymeric liquids have received growing attention for the understanding of the physics of the processes, for the design of the equipments and for control purposes. The theoretical foundation of this software is provided by principles of continuum mechanics, together with improved numerical methods for the solution of the mathematical equations and by the use of pertinent constitutive equations for the description of the theological behaviour of molten polymers. Most flows encountered in processing or experimental problems are characterized by complex kinematics involving different geometries combining shear and elongation, different time dependences and amplitudes of the deformations. Moreover, the behaviour of polymeric materials under such conditions exhibits a large variety of specific features such as the shear rate dependence of the viscosity, appearance of norma| stresses, memory effects and high resistance in elongation. Among the large m~nber of existing models, only a relatively few equations can predict the variety of phenomena encountered in the flow of polymer melts and can give a fair description of the physics involved in the theological

142 behaviour of these materials, both from a microscopic and a macroscopic point of view. From the physicist's and chemist's points of view, the ideal constitutive equation should be able to describe the behaviour of polymers in any flow situation without any adjustable parsmeter. Linear and non linear viscoelastic characteristics of the material should be theoretically described from molecular dynamics, knowing the basic properties of the polymer molecules, from the monomeric unit to the molecular weight distribution as obtained from the polymerization process. Many advances have been performed in this sense in the past few years (see Section 1.5), however the complexity of the equations that might be obtained to take into account these various features may lead to numerical difficulties. Indeed, the computation of complex flows, such as those encountered in typical processing conditions, sets its own practical requirements on the constitutive equation, especially when reasonable computation times and memory requirements are expected. The numerical properties of the equation generally do not match those of the physicist. In this sense, for example, the use of a large number of relaxation modes or of a continuous spectrum to describe the memory function is time and memory consuming and the economic spectrum using only a few contributions, though unsatisfactory from the molecular dynomics point of view, remains the rule. Thus, this constitutive equation is bound to be replaced by an unsatisfactory but easy to handle model equation which involves a minimum of violation of basic principles of material physics. This equation will necessarily contain a few adjustable material parameters, which have to be easy to determine in a limited number of well defined flow experiments. Obviously, the minimum requirement that could reconcile the two stand points should be the ability of the model equation to describe properly the response of the polymer to simple viscometric flows (simple shear, uniaxial, biaxial and planar extension) in which quite perfectly controlled conditions can be obtained. From this, it can be hoped that the situation might be at its best in the combined complex flow. The aim of this section is to perform comparisons between the predictions of some constitutive equations and experimental results in simple shear and uniaxial elongation on three polyethylenes. In addition, this is expected to provide well-defined sets of material parameters to be used in the model equations for the computation of complex flows.

143 L2.Network theories for polymer melts and related models. Among the various approaches in use for the depiction of the interactions of the polymer molecules in the melt, these being known to be at the origin of the observed theological behaviour, the network theories enable the building of reasonable models t h a t fulfill the previous requirements for the sake of simplicity. These theories are based on the classical theories of rubber elasticity of maeromoleeular solids, wherein permanent chemical crosslinks connect segments of molecules, forcing them to move together. This central idea can be applied to polymeric liquids. However in this case, the interactions between molecules are assumed to be localized at junctions and are supposed to be temporary. Whatever their nature, physical or topological, these crosslinks are continually created and destroyed but, at any time, they ensure sufficient connectivity between the molecules to give rise to a certain level of cooperative motion. The stress is considered to be the sum of the contributions of segments between junctions that are still existing at the present time. By the way, these segments, t h a t were created in the past, may be of different ages and complexities. This time dependence is generally described by a relaxation spectrum t h a t gives rise to the linear rheologieal behaviour. Strain dependence, at the origin of nonlinearity, can be described either by changing the motion of the network relative to the continuum or by special rates of creation and loss of junctions. Owing to their relatively fair tractability and because they retain some physical consistency, network models are widely used in computer simulation of the flow of polymer melts. Thus, the attention of the present article is focused on constitutive equations of this type. l~.Integral and differential forms of the models. At this point, the question of the use of either an integral or a differential equation arises. Integral forms are closer to those obtained by the recent molecular dynamics concepts for entangled polymer melts. Unfortunately, their use requires the knowledge of the Finger strain tensor in complex kinematics, which together with the fluid memory, involve the description of the material history. This, in turn, sets the difficult task of particle tracking. Though it has been difficult to cope with, alternative descriptions (Protean

144 coordinates) and new ideas (such as those of the streAm-tube, see Section III-2) enlighten the subject and bring new hope for this kind of equation. In this sense, differential equations appear more tractable since they do not require particle tracking. Indeed, the solution of the coupled equations of mass, momentum and energy balance including the material equation, properly described on a suitable finite element mesh, theoretically provides the material lines. Nevertheless, the correct description of the basic experiments ot~en requires the use of strong nonlinear terms. Such improvements may be unsatisfying from the numerical point of view since they can lead to stiff systems of nonlinear equations and to many convergence related problems. Considering these previous remarks, two network models, thought to be representative of each class of equation, have been investigated, namely the Wagner model and the Phan-Thien Tanner model, 1.4.Experimental validation of network model~ P a r t 2 presents a summary of the theoretical considerations and basic assumptions t h a t lead to the model equations. Part 3 discusses some experimental aspects and focuses on the measurements in various shear and uniaxial elongational flow situations. Part 4 and 5 are devoted to the comparisons between experimental and predicted rheological functions. Problems encountered in the choice of correct sets of parameters or related to the use of each type of equation will be discussed in view of discrepancies between model and data. 2~THEORETICAL ASPECTS 2.1.The basic integral and differential non-linear constitutive equation~ 2.1.1.The linear M~xwell model and its limits.

Constitutive equations of the Maxwell-Wiechert type have received a lot of attention as far as their ability to describe the linear viscoelastic behaviour of polymer melts is concerned. From a phenomenological point of view [1-4], these equations can be easily understood and derived using the multiple springdashpot mechanical analogy leading to the linear equation :

145

~i(t) + ~ ~

= Th~ and ~=(t) = .~ ~(t)

(1)

1

where

~i(t) is the contribution of the ith Maxwellian assembly (spring and

dashpot in series) to the extra-stress tensor ~(t), d ~-~is the tensor time derivative for small displacements, ~ = Vu + (Vu) t is the rate of strain tensor (u being the fluid velocity), is the relaxation time of the ith element, Tli = gi ~ is the viscosity contribution of the ith element, gi is the modulus contribution of the ith element. This differential form can be integrated to give the integral form of the model which can also be derived from the Boltzman superposition principle using the concept of fading memory of viscoelastic liquids: t

~(t) =- j re(t- t') dt(t') dt'

(2a)

.-OO

or

t ~=(t)= j G ( t - t ' ) ~ dt'

(2b)

-00

where

Th t m(t) = ~ ~.2 exp{-~.} is the memory function, Th t din(t) G(t )= ~ ~. exp{-~.} =- dt is the relaxation modulus, dt(t') is the infinitesimal strain tensor (t being the reference of the

deformation, so that the strain at time t' is relative to that at time t ). Satisfactory agreement is achieved from these equations for depiction of the main features of linear viscoelastic properties that can be obtained with the experimental tools, either in transient or in oscillatory rheometry. These equations are all the more attractive in that similar mathematical forms can also be obtained from molecular considerations for the description of

146 the flow behaviour of non-entangled [5] and entangled [6-10] melts at least in the case of narrow molecular fractions, so that any parameter in equations (1) or (2) becomes physically meaningful. However, the latter approach leads to complex relaxation spectra with a large number of Maxwellian modes (and maybe even more complex if polydispersity and branching are taken into account). Recognizing that only one or two modes per decade are generally sufficient to get a proper description of the linear viscoelastic behaviour [11], such an economic spectrum can be numerically adjusted [11-17] and is often used, while keeping in mind that, in this case, one loses the physical meaning of such modes. Though a simple Maxwell model in the form of equations (1) and (2) is powerful to describe the linear viscoelastic behaviour of polymer melts, it can do nothing more than what it is made for, that is to describe mechanical deformations involving only infinitesimal deformations or small perturbations of molecules towards their equilibrium state. But, as soon as finite deformations are concerned, which are typically those encountered in processing operations on polymers, these equations fail. For example, the steady state shear and elongational viscosities remain constant throughout the entire rate of strain range, normal stresses are not predicted. 2.1.?~The ~ e

and UCM models.

One major discrepancy of the previous model can be attributed to the use of the infinitesimal strain tensor and to derivatives restricted to time changes. Indeed, in the case of large deformations, one has to refer to finite strain tensors, such as the Finger c t ' l ( t ') or Cauchy Ct(t') strain tensors (t being the present time as a reference of the deformation), and to derivatives involving time and space changes known as convected derivatives. Making these modifications, the integral form of the equation of state becomes: t ~(t) =-p'~ + ~ r e ( t - t ' ) C t l ( t ') d t ' = - p ' ~ + ~=(t) -OO

where

~ is the total stress tensor, 5=is the unit tensor,

(3)

147

p' is an arbitrary isotropic term. For an incompressible liquid, because of the arbitrary term, normal stresses are known only up to a constant using the constitutive equation. This equation was proposed by Lodge [18, 19] in the case of entangled polymer molecules, considering that the response of polymers to flow is that of a temporary network of junctions. The strands of the network can be of different topology (i) but they are considered as Gaussian springs that deform like the macroscopic continuum. This latter a s s u m p t i o n of affine motion m e a n s t h a t any s t r a n d end-to-end vector, initially coincident with a macroscopic vector embedded in the continuum, remains parallel to and of equal length with it during deformation. The s t r a n d s are continuously destroyed and rebuilt by thermal effects only. Assuming t h a t the rates of creation and destruction of the ith segment are constant, an equation for the memory function similar to that of equation (2) can be written, with viscosity contributions Tli connected to the stiffness of any ith spring and to its rate of creation, and relaxation time ~i connected to the survival probability of the ith junction. This formulation provides the way to cope with the economic spectrum discussed previously as a crude physical description of the entangled melt. The Lodge equation can also be obtained in a differential form known as the Upper Convected Maxwell equation (UCM): ~i(t) + ~ ~i(t) = Tli ~

and ~=(t) = .~ ~(t)

(4)

1

where ~i(t) is the upper convected derivative of the contribution of the ith mode to the extra stress tensor ~=(t). It should be noted that this is the exact derivation of equation (3). These models are usually referred to quasi-linear models and display qualitatively correct predictions of typical phenomena of elongational flows such as the occurrence of the strain-hardening effect in transient extension. Nevertheless the predicted elongational viscosity is never bounded in the long time range and a steady state value can only be expected for small elongation rates. Moreover, the shear behaviour r e m a i n s unrealistic as compared to the experiment, especially because of constant predicted viscosity and first normal

148 stress coefficient. The cure of these discrepancies either requires much more complexity in the formulation of the constitutive equation or an additional attention to some characteristics of the viscoelastic material that might only be displayed in nonlinear flows. 2.2.An i n t e g r a l constitutive equation: the Wagner modeL

2.2.1.The general K-BKZ model. Additional complexity can be brought to the constitutive equation in its integral form. Indeed, the idea of rubber elasticity that is inherent to the Lodge model has been generalized by Kayes, Bernstein, Kearsley and Zapas [20-23] in a large class of constitutive equations. In a perfect body, the strain energy W may be linked to strain and stress by:

-

8W be

(5)

1 where e= is a diagonal strain tensor (Hencky strain tensor) such as ~== ~ In C "1.

This can also be rewritten in terms of the Cauchy and Finger strain tensors as:

1: =2~IW1 C -t 8W =

= "~I2 C=}

(6)

Since W is a scalar value, it depends on scalar characteristics of the strain tensors, namely on the invariants I1 and I2 defined as: I1 = tr[C 1]

1

I2 = ~ {(tr[C

(7a)

-1])2

- tr[C2]} = tr[C]

(7b)

In r u b b e r and viscoelastic fluids, these two quantities are sufficient since, when incompressibility is taken into account, I3 = 1. For viscoelastic fluids, both strain energy and stress can be assumed to depend on the strain history through the strain invariants:

149

t W = ~ u ( t - t', I1, I2) dt'

(8)

-OO

and thus:

~(t) = ~ 2 {

C t l ( t ' ) " ~22 Ct(t')} dt'

(9)

-00

or

t ~(t) = ~ {Ml(t- t', I1, I2) c t l ( t ' ) - M2(t- t', I1, I2) Ct(t')} dt'

(10)

--OO

where M l ( t - t', I1, I2) and M2(t- t', I1, I2) are m a t e r i a l functions. The K-BKZ has the interesting property being t h e r m o d y n a m i c a l l y consistent because one can theoretically derive two material functions which depend on a single potential function u in the general form: Mi(t- t', I1 I2) = 2 8u(t- t', I1, I2) ~Ii

(11)

'

The Lodge model is a special case of this class of models, where the material functions are selected as: M l ( t - t', I1, I2) = m ( t - t')

(12a)

M2(t- t', I1, I2) = 0

(12b)

1 u ( t - t', I1, I2) = ~ m ( t - t') (I1- 3)

(13)

However, although it has some t h e r m o d y n a m i c consistency, the l a t t e r model failsto describe the non linear viscoelastic b e h a v i o u r properties, especially in shear, wherein the shear-thinning behaviour of the viscosity and of the normal s t r e s s coefficients are not predicted. As a c o n s e q u e n c e , m o r e complex

150 n o n l i n e a r functions have to be introduced to get a proper description of the observed behaviour. 22..2.Time-strain separability. Some experimental features might simplify the problem. Considering t h a t in step shear strain of constant ,mplitude 7 starting from the material at rest, the K-BKZ model leads to the following equation for the shear stress: t 9(t, 7) = ~ {Mi(t - t', Ii, I2) - M2(t- t', Ii, I2)} 7 dt' 0

(14)

and since, in this case: Ii=I2=3+

T2

(15)

one can write" t 9(t, 7) = ] M(t- t', Y)7 dt' 0

(16)

The nonlinear relaxation modulus is then obtained as:

G(t, T) = Z(t, 7) 7

(17)

As a l r e a d y mentioned by various authors [24, 25], it is found experimentally t h a t at various shear strains, for polydisperse polymers, the logarithmic plots of G ( t , 7) v e r s u s time are only shifted vertically. This indicates t h a t the nonlinear relaxation modulus might be factorizable: G(t, 7) = G(t ) h(7)

(18)

where h is a strain-dependent function, between zero a n d unity, called the d a m p i n g function. This t i m e - s t r a i n s e p a r a b i l i t y seems to hold in the experimental range of shear strains. From a theoretical point of view, this can only be achieved if u ( t - t', Ii, I2) is also factorizable:

151

u(t - t', I1, I2) = m ( t - t') U(I1, I2)

(19)

As a consequence, the K-BKZ model is now written in the less general form of the factorizable K-BKZ model: t 8U 8U ~(t) = ~ 2 m ( t - t') {~-~1ct'l(t') " ~ 2 Ct(t')} dt'

(20)

-00

or

t ~(t) = ~ re(t- t') {hl(I1, I2) c t l ( t ' ) - h2(I1, I2) Ct(t')} dt'

(21)

-OO

It is worth mentioning that the strain function is not t e m p e r a t u r e dependent and t h a t the influence of temperature is only applied on the memory function or relaxation modulus through the shortening of the relaxation times with increasing t e m p e r a t u r e s . 2.2.3.The Wagner m o d e l Since second normal stresses are generally difficult to obtain from the experimental point of view, it may seem attractive to cancel the Cauchy term of the K-BKZ equation setting h2(I1, I2) = 0 and to find a suitable material function

hl(I1, I2). Wagner [26] wrote such an equation in the form: t =(t) 9 = ~ m ( t - t') hl(I1, I2) C t l ( t ')

(22)

-OO

Cancellation of the Cauchy term may bring some discrepancies, the more evident one being that, whatever h is, it leads to a zero second normal stress difference. A more subtle one concerns the loss of the t h e r m o d y n a m i c consistency of the model. Indeed, it is not possible to find any potential function in the form U(I1, I2) with h2(I1, I2) = 0 unless hi only depends on I1. As mentioned by Larson [27, 28], this can induce violation of the second principle in complex flows such as those encountered in processing conditions.

152 22~4.The g e n e r a l i z e d i n v a r i a n t The form of the function h can be described from shear experiments. Indeed, since in these experiments" I1 = I2 = 3 + 7t2(t')

(23)

The mathematical form of the function can be derived simply from a fit of the experimental h(T) as obtained in step shear strain for example. However, the problem is further complicated if one now takes into account flows where the two invariants differ from each other as, for example, in uniaxial elongational flows where: I1 = exp{2et(t')} + 2 exp{-et(t')}

(24a)

I2 = exp{-2et(t')} + 2 exp{et(t')}

(24b)

In order to conciliate these features, Wagner [29, 30] proposes the use of a generalized invariant: I = ~ I1 + (1-~) I2

(25)

which enables the derivation of a single form of the damping function, since in s h e a r flows I = I1 = I2. It should be pointed out t h a t one m a y find other generalized invariants which satisfy this condition. These have been proposed by various authors but are less frequently used, for example: I = (I1)a(I2)1~

(26)

In any case, the new parometer of the generalized invariant has now to be obtained from additional experiments in elongational flows. But the Wagner equation can now be written in the unified form:

~=(t) =

t ] m ( t - t') h(I) c t l ( t ') -OO

(27)

153 In terms of network description, Wagner considers t h a t the d a m p i n g function may reflect an additional process of destruction of network junctions by strain effects, as described by the generalized invariant, thus involving some peculiarities of the flow such as its geometry and the s t r a i n intensity. As regards the rate of creation of network junctions, it is a s s u m e d to remain constant as in the Lodge model. 2~2.&Interpretation a n d mathemotical forms of the d a m p i n g functiom In its form, the model of equation (27) is very useful since using a proper damping function enables a correct description of various shear or uniaxial elongational basic experiments. Going back to the Lodge model, Wagner keeps the assumption of constant creation rate (connected with Tli for the ith segment in the memory function) but assumes t h a t the loss probability of junctions is now a combination of two independent mechanisms. One is related to Brownian motion as in the case of the Lodge model and depends on the time elapsed between the creation of an entanglement and the present time (t- t'), the survival probability being related to 1/ki. The second reflects the network breaking by deformation and depends on the kinematics of the flow and not on the segment configuration, it is expressed as a survival probability between times t' and t. Since the processes are assumed to be independent, the total loss probability is j u s t the sum of the former and this leads to a separable nonlinear memory function : M(t- t') = m ( t - t') h(I).

(28)

Wagner proposes a single exponential form of the damping function: h(I) = exp(-n~] I- 3).

(29)

The exponential form is interesting because, in shear, the response of the model can be analytically derived. However because of the exponential, it decays very rapidly, even at low deformation and therefore it cannot take into account the linear viscoelastic domain, which is sometimes found to extend to relatively high values of the strain (typically 0.5). Another interesting form was used by Papanastasiou and al. [31]:

154

1 h(I)=l+a(i.3)

"

(30)

Using a factorizable K-BKZ equation (21), Wagner and Demarmels [32, 33] showed that an equation of the damping function such as 9 h(I1, I2)=

1 1 + a~](I1 : 3)(I2 - 3)

(31)

m a y be suitable for s h e a r and uniaxial extensional flows. Though it is not written in term of a generalized invariant, it degenerates to equation (30) in shear. The i n t e r e s t of equation (30) also lies in its s i m i l a r i t y with a p p r o x i m a t i o n s of n o n l i n e a r functions obtained from the Doi-Edwards constitutive equation [8] for the reptation theory, at least in shear. Indeed, the later authors have developed a simplified equation in the form of the K-BKZ model considering the "independent alignment" ass,lmption, which states that the strands contract back after the strain is imposed and before relaxation occurs, so that they are only oriented and not deformed. Currie [34] found an accurate approximation for the related potential function U(I1, I2) which, in shear, leads to:

h(7) =

5 N~4~+ 25 + 10 (~2+ 2)~J4r~+ 25 + (4r~+ 25)

(32)

Equation (30) is an approximation of equation (32) in shear. Larson [35] has further simplified the Currie potential to obtain U(I1) and he derived h(7) in the form of equation (30) ~ i t h a = 0.2. In terms of network analogy, the damping function may be viewed as the expression of the retraction of the strands as compared to the continuum. The Lodge model thus corresponds to no retraction (affine deformation, a=0 in equation (30)), the Doi-Edwards equation corresponds to complete retraction (a=0.2), whereas incomplete retraction makes the damping function more softly decreasing (0 < a < 0.2). In the later cases, the deformation is non-affine since there is a difference between t h a t of the continuum and t h a t of the network strands. Wagner [33] showed that the Doi Edwards strain function

155

exaggerates the strain dependence and that obviously complete retraction is not consistent with the data. Another form which, due to a third parameter, enables a slightly better approximation of equation (32) and of the experimental data was used by Soskey and Winter [36] in the form: 1

h(I) = 1 + a (I- 3)b~2

(33)

Parameters a (and b) can be associated with the completion of the retraction process together with the strain amplitude. 2,2.6.Integral t e m p o r a r y n e t w o r k models and molecular theories. Wagner and Schaeffer [37-39] made an interesting attempt to reconcile the different aspects of the temporary junction network model together with the Doi-Edwards model. They proposed a simple picture of the effects of large deformation on the stress in terms of a slip links model. Assuming t h a t the entanglements can be thought as small rings through which the chain may reptate freely, in addition to equilibration and reptation, two deformation processes are assumed to give rise to the observed nonlinear behaviour after a step strain. The first process is connected to equilibration of the monomeric units between the entanglements by a slippage of the chains and is described by a normalized slip function S, giving the number of monomers in a deformed s t r a n d relative to equilibrium. The second process is related to a loss of junctions at the chain ends or along the chains by constraint release described by a normalized disentanglement function D giving the mlmber of strands for a deformed chain relative to equilibrium. These functions are connected to the tube dimensions (relative length of the strands or tube segments u' and tube diameter a). Since the number of monomers on a chain is balanced, at equilibrium, the average over the configuration space of their product is unity : = 1. Calculating the stress with various assumptions on the functions leads to different types of equations with different strain measures : -No slip and no disentanglement (D = 1 and S = 1) leads to the Lodge model. -Isotropic slip and disentanglement (D = , S = <S>, D . S = = 1) leads to the Wagner model with h(I1, I2) = D 2. -Slip related to relative strand extension but constant tube diameter (S = u', the molecular tension in a deformed chain is equal to its equilil~rium value)

156 and anisotropic or isotropic disentanglement (D. S = 1 or D . <S> = 1) provides the Doi-Edwards model with or without the " i n d e p e n d e n t alignment" assumption. -Slip related to relative strand extension but constant tube volume (S = u 'y2, the molecular tension in a deformed chain depends on the individual segmental stretch) and anisotropic d i s e n t a n g l e m e n t (D . S = 1) slightly improves the predictions [40]. -Wagner and Schaeffer ass~lmed t h a t the tube diameter is a function of the average stretch ], the molecular tension in a deformed chain depends on the average segmental stretch) and anisotropic disentanglement (D . S = 1). The function f can be theoretically derived from the experimental damping function. 2 ~ 7 J r r e v e r s i b i l i t y assumptiom As mentioned by several authors, there is experimental evidence that the process of loss of junctions, whatever its nature and whatever the domping function, may be irreversible. This led Wagner and Stephenson [41, 42] to consider t h a t their original equation is only valid in experiments wherein the d e f o r m a t i o n is monotonicallly increasing. To t a k e into a c c o u n t the irreversibility of the loss of junctions, which means t h a t these are never rebuilt in a decreasing deformation following an increasing one, they suggested the use of a functional r a t h e r than a damping function. So that, in the original model, the damping function should be replaced by: H(II(t,t'), I 2(t,t ' )) = ~ f m tC. _= t t, h(Ii(t',t'), I 2(t, " t ' ))

(34)

In a monotonically increasing deformation, H(I1, I2) = h(I1, I2).

2.3.A differential constitutive equation: the Phan-Thien T a n n e r modeL 2~3.12VIodifications of the UCM m o d e l Many improvements or modifications to the UCM model can be found in the literature. These can lead to various classes of constitutive equations keeping the differential nature of the equation [2, 3, 35]. As pointed out by Larson [43], a systematic classification of these can be performed by rewritting the UCM model as:

157

~(t)- ~

1

(35)

_

The various changes that may be carried out can be either on the convected derivative or in the right term of equation (35) or both; these imply the removal of some assumptions of the initial model. Such a possible modification, that was claimed to give a correct description of the essential phenomena of the nonlinear viscoelastic behaviour of polymer melts, is that proposed by Phan Thien and Tanner [44-46] involving the use of a special convected derivative and special kinetics of the junction. ~2~on ~ m o t i o n . The G o r d o n - S c h o w a l ~ derivative. The first kind of modification to the UCM model that may be conceivable is t h a t of the convected derivative. This leads one to consider that the motion of the network junctions is no more t h a t of the continuum and thus, the affine assumption of the Lodge model is removed. Among the various possibilities, P h a n Thien and Tanner suggested the use of the Gordon-Schowalter derivative [47], which is a linear combination of the upper- and lower-convected derivatives, instead of the upper-convected derivative: ~(t) = (1 - ~) a ~i(t) + ~a

(t)

0 ~.~mL 104 I

m -F,4

""'""-.

~u=L~-u~

o~ L

~ L

~r "~

u~

10 2

[

1O- 3

.................

4/

1 O- 2

'

1 O- 1

................................... 10 ~

Shear" or Elong.

101

Rate

10 2

10 3

[s-l]

Figure 14: Steady state functions for LLD at 160~ (experimental d a t a (o): elongational viscosity, (Q): shear viscosity, (A): first normal stress difference and fit (--) e = 0.105, (- -): e = 0.7)

It is thus impossible to find a single value that enables correct description of both shear and elongational data. This m a y be understood considering the efficiency of the Y function in describing the shortening of the junction lifetimes. In the model, this shortening is all the more important since the stress magnitude is higher. Since it is generally observed that materials which exhibit the highest stress in elongation (elongation thickening) also show the opposite trend in shear (shear thinning), the weighting by the function can hardly be achieved in any coherent way with a single value of e in different flow

geometries. Once more, as in the Johnson-Segalman equation, this sets an important limitation for the easy handling of such an equation. 5 ~ C o m b i n a t i o n of the two modifications-Experimental vah'dation of the P b a n Thien T a n n e r model The

original

Phan

Thien

Tanner

equation

was

written

using

simultaneously both modifications: Gordon Schowalter derivative and segment kinetics term. The segment kinetics term (exponential form) enables a more

186

realistic description of the steady elongational behaviour, giving rise to a bounded viscosity in the long time range. The Gordon Schowalter derivative has its major influence on the shear properties and additionally predicts a second normal stress difference as in the case of the Johnson-Segalman model, equation (53). Unfortunately it also introduces, conversely, the infringement of the Lodge Meissner rule. Considering the previous remarks, one must keep in mind the following important points. The smaller the value of a, the lower the deviation to the Lodge-Meissner rule. However, in this case the flow behaviour of the model is primarily described by e and the simultaneous description of shear and elongational data using a single value of this p a r a m e t e r is impossible. The smaller the value of e, the higher the value of the steady elongational viscosity can be. However, in this case the shear flow behaviour of the model is described by a so that the violation of the Lodge Meissner rule may become important. The predictions are then very close to those of the JohnsonS e g a l m a n model with the associated discrepancies such as spurious oscillations in transient shear. The determination of a couple of values (a,e) is then bound to be a compromise obtained from a simultaneous fit of the elongational and shear functions (Table 6). Table 6: Parameter a and e for the various materials Material

a

E ,

,

. . ,

.

.

HD LD LLD

,

.

9

.

,

0.50 0.15 0.35

,

,.

,

,

,,==

0.050 0.025 0.060

Figures 15 to 18 show the predictions of the model for LD at 160~ in steady state and some transient flows in shear and uniaxial elongation.

187

~ ~ I06 ~'--'

~

o

o

i0 s

U

>

~

10 4

o~ E

~Zo~b103. "~

10 2

I0 -3

///:" . . . . . . .

.

I

~

.

I0 -2

.

.

.

I

I0 -I

. I-ill

. . . . . . . .

I

. . . . . . . .

....................

I0 ~

I0 x

I(0 2

103

Shear or Elong. Rate [s - I ]

Figure 15: Steady state functions for LD at 160~ (experimental and calculated). (o): elongational viscosity, (u): shear viscosity, (A): first normal stress difference.

10 5

-

U~

.

EIO

130

131"100rl

o~

10 3

I0 - I

. . . . . . . .

j

10 ~

. . . . . . . .

,

101

. . . . . . . .

,

10 2

Time [ s ]

Figure 16: Transient shear viscosity for LD at 160~ calculated). (o): 0.2 s -1, (u): 0.5 s -1, (A): 1 s -1.

(experimental and

188

10 6 o

O0 0

105

L p-g m E (. 0 Z L o,-g LL

10 4

103

10-1

10~

Time [s]

101

102

F i g u r e 17: T r a n s i e n t p r i m a r y stress coetticient for LD a t 160~ a n d calculated). (o): 0.2 s -1, (~): 0.5 s -1, (A): 1 s -1.

10 6

(experimental

-

105 -,-4

-a=4

>

10 4

o W

103 10-2

.

.

.

.

.

.

,,I

10 -1

9

,

. . . . . .

i

. . . . . . . .

10 o

,i

i

i

l

101

.....

i

10 2

Time [ s ]

F i g u r e 18: T r a n s i e n t elongational viscosity for LD at 160~ calculated). (o): 0.05 s -1, (Q): 0.5 s -1, (A): 1 s -1, (0): 2 s -1.

(experimental and

189

5.4.Conclusion. The ability of the Phan Thien Tanner equation and related models for the prediction of data in shear and elongation has been investigated. Attention has been focused on special simplified cases of the original equation which enable

the understanding of the influence of each parameter. The use of a single parameter equation, removing the affine assumption of the U C M model by replacement of the upper-convected derivative by the Gordon-Schowalter derivative, is the case of the Johnson-Segalman model. This kind of modification does not significantly improve the elongational prediction towards the U C M equation. Moreover, in shear, though the improvement is obvious, discrepancies remain, especially concerning the nonuniqueness of the slip parameter for tangential and normal stresses. The change of kinetics of the junctions also leads to a single parameter equation in the form of the original PTT equation but using the upper-convected derivative. Only the exponential form of the stress term gives a realistic description of the steady elongational behaviour in the long time range. Though this is shown to improve the behaviour in elongation, this was conversely found to be contradictory with a better description of the trend in simple shear because of opposite requirements on the value of the parameters. Indeed, the parameter that controls the kinetics promotes both an elongation thickening behaviour together with a shear-thinning trend which is in contradiction with experimental data on LD for example. At least, using the complete Phan Thien Tanner equation, with non-affine motion and modified kinetics enables a correct description of the data in shear and in elongation. However, the parameters t h a t can be determined for this model are bound to be some compromise. This is n e c e s s a r y in order to minimize the deviation to the Lodge-Meissner rule, due to the use of the Gordon-Schowalter derivative. This is also r e q u i r e d to give a d e q u a t e description of both the shear and uniaxial elongational behaviour. Additional undesirable phenomena in some flows have also been pointed out such as oscillations in transient flows. At least, it is worth noticing that the Phan Thien Tanner model is, in its m a t h e m a t i c a l form, a non-separable equation. However, it has been pointed out t h a t , for some special forms of the r e l a x a t i o n spectrum, a p p a r e n t separability may be displayed [61].

190 6.CONCLUSION Two different constitutive equations, namely the Wagner model and the P h a n Thien Tanner model, both based on network theories, have been investigated as far as their response to simple shear flow and uniaxial elongational flow is concerned. This work was primarily devoted to the determination of representative sets of parameters, that enable a correct description of the experimental data for three polyethylenes, to be used in n u m e r i c a l calculation in complex flows. Additionally, a d v a n t a g e s and problems related to the use of these equations have been reviewed. Both these models find their basis in network theories. The stress, as a response to flow, is assumed to find its origin in the existence of a temporary network of junctions that may be destroyed by both time and strain effects. Though the physics of time effects might be complex, it is supposed to be correctly described by a generalized Maxwell model. This enables the recovery of a representative discrete time spectrum which can be easily calculated from experiments in linear viscoelasticity. Though the Wagner and Phan Thien Tanner equations seem to give adequate description of the observed behaviour either in shear or in uniaxial elongation, it is worth mentioning some peculiarities and key points that should keep the attention of the user to avoid misleading conclusions. These constitutive equations differ in their mathematical form: the Wagner equation is an integral equation whereas the Phan Thien Tanner model is a differential one. This induces important differences in the way they might be treated for calculations in complex flows, since integrals will require particle tracking whereas differential equations will not. These numerical t r e a t m e n t s are generally mutually exclusive since, in the general case, the problem of correspondence between integral and differential forms is not solved. Attempts at finding such correspondences may be found in various papers by Larson [62, 63] especially concerning the Wagner model and the Phan Thien Tanner equation with upper-convected derivative. On the other hand, integral forms are closer to the results of our knowledge of molecular dynamics in entangled polymers and hybrid theories combining

191 simplified molecular models and temporary network equations are worth thinking over. The Wagner equation finds its theoretical basis in the derivation of the more general K-BKZ equation. Unfortunately, it loses p a r t of its original thermodynamic consistency since, for simplification purposes, only the Finger strain measure is taken into account. Doing so, it is no more derivable from any potential function and additionally it does not predict second normal stress differences any more. The equation leads to the definition of a time and strain-dependent memory function which can be further factorized into a time-dependent part (the linear memory function) and a strain-dependent damping function. Though on one hand, there is some experimental evidence for this in limited time ranges, on the other hand, a few experiments might question this strong hypothesis since, for example, the damping function obtained from step shear rate data is found to be different from that in step shear strain. The construction of a single mathematical form of the damping function in shear and uniaxial elongational flows requires the use of a generalized invariant, which includes the effect of both strain invariants I1 and I2 through a proper combination of them. The simplest one being a linear combination can be used in various equations for the damping function, including a limited number of adjustable parameters. Including these features, the Wagner model can give a proper description of experiments in shear and in uniaxial elongation for increasing deformations. When deformation is non-increasing, since the damping function reflects the loss of junctions under the influence of strain, and since it should obviously be an irreversible process, a functional damping term has to be introduced. Nevertheless, this key point for any use in complex flow calculations has to be improved. In its general form, the Phan Thien Tanner equation includes two different contributions of strain to the loss of network junctions, through the use of a particular convected derivative which materializes some slip of the junctions and through the use of stress-dependent rates of creation and destruction of junctions. The use of the Gordon-Schowalter derivative brings some improvement in shear and a second normal stress is predicted, whereas the

192 influence of the kinetics through the trace of the stress tensor is much more important in elongation. Unfortunately, the use of the Gordon-Schowalter derivative brings large discrepancies, especially as far as material objectivity is concerned. Indeed, using it, the principal axes of strain and stress do not rotate together during shear flows and this induces the violation of the Lodge Meissner rule. Consequently, the slip parameter of the derivative is found to be different for tangential and normal stress functions. This becomes evident in the limiting case of the Johnson-Segalman model which, for representative parameters, is found to be a good approximation of the Phan Thien Tanner model in shear. Moreover, this kind of derivative induces spurious oscillations for transient rheological functions. One may try to avoid the problem by the use of the upper-convected derivative, which ensures the coincidence of the principal axes of stress and strain. But doing that, it appears that any kinetics based on the stress amplitude is improper, since materials which exhibits thickening behaviour in elongation are, to the contrary, shear-thinning. Consequently no unique dependence can be expected for these two kinematics. The determination of a single set of parameters in various flows in then bound to be a compromise. Table 7 gives a summary of qualitative performances and problems encountered for simple shear and uniaxial elongational flows, using the Wagner and the Phan Thien Tanner equations or more simple models as special cases of the former. Additional information may also be found in papers by Tanner [46, 64]. All equations presented hereafter can be cast in the form of a linear Maxwell model in the small strain limit and therefore are suitable for the description of results of the linear viscoelasticity in the terminal zone of polymer melts.

193 Table 7: Wagner and Phan Thien Tanner equations- Problems in simple shear and uniaxial extension. Model Comments cD

~D

UCM-Lodge Eqs.(3) or (4) o

Constant viscosity and first normal stress difference. Second normal stress difference is zero. No overshoot and linear limits in transient stress growth. Linear relaxation modulus in step shear strain. Unbounded transient viscosity at high rates. Strain hardening at short time. Linear results or infinite value for steady state viscosity. Second normal stress difference is zero.

Wagner Eqs.(33 ) and

(25)

JohnsonSegalman Eq.(49)

Inaccuracy on the generalized invariant parameter. o p==r

Exaggerates shear-thinning. Lodge Meissner rule unsatisfied (2 slip parameters). Oscillations in transient stress growth. Negative relaxation modulus in large step shear strain. bb o ~=~

Phan Thien Tanner with UCD

c~ cD

o

c~

Phan Thien Tanner Eq.(39) o

Unbounded transient viscosity at high rates. Linear results or infinite value for steady state viscosity. Non-separable equation.

Linear form of the junction kinetics unsuitable. Parameter of the junction kinetics differs from shear. Non-separable equation. Lodge Meissner rule unsatisfied. Oscillations in transient stress growth. Linear form of the junction kinetics unsuitable. Compromise is necessary for the parameters.

194 R~'~C~.

1. 2. 3.

4. 5. 6.

7.

8.

9. 10. 11.

12.

13.

J.D.Ferry,~Viscoelastic Properties of Polymers ~, 3rd edition, JohnWiley &Sons, 1980. R.B.Bird, R.C.Armstrong, O.Hassager,~Dynamics of Polymeric Liquids" Vol.1, Fluid Mechanics, 2nd edition, John Wiley & Sons, 1987. R.B.Bird, C.F.Curtiss, R.C.Armstrong, O.Hassager,~Dynamics of Polymeric Liquids ~, Vol.2, Kinetic Theory, 2nd edition, John Wiley & Sons, 1987. N.W.Tschoegl, ~The Phenomenological Theory of Linear Viscoelastic Behavior- An Introduction ~, Springer Verlag, 1989. P.E.Rouse, A theory of linear viscoelastic properties of dilute solutions of coiling polymers, J. Chem. Phys. 21 (1953), 1272-1280. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 1: Brownian motion in the equilibrium state, J. Chem. Soc, Faraday Trans / / 74 (1978), 1789-1801. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 2: Molecular motion under flow, J. Chem. Soc, Faraday Trans H 7..44(1978), 1802-1817. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 3: The constitutive equation, J. Chem. Soc, Faraday Trans H 74 (1978), 18181832. M.Doi, S.F.Edwards, Dynamics of concentrated polymer systems- Part 4: Rheological properties, J. Chem. Soc, Faraday Trans H 75 (1979), 38-54. M.Doi, S.F.Edwards, ~The Theory of Polymer Dynamics", Clarendon Press, 1986. M.Baumgaertel, H.H.Winter, Determination of discrete relaxation and retardation time spectra from dynamic mechanical data, Rheol. Acta 28 (1989), 511-519. M.Baumgaertel, H.H.Winter, Interrelation between continuous and discrete relaxation time spectra, J. Non-Newt. Fluid Mech. 4.~4(1992), 1536. M.Baumgaertel, A.Schausberger, H.H.Winter, The relaxation of polymers with linear flexible chains of uniform length, Rheol. Acta 29 (1990), 4(D-408.

195

14.

15. 16.

17.

18.

19. 20. 21.

22. 23. 40

25. 26.

27. 28.

J.Honerkamp, Ill-posed problems in rheology, Rheol. Acta 28 (1989), 363371. J.Honerkamp, J.Weese, Determination of the relaxation spectrum by a regularization method, Macromolecules 22 (1989), 4372-4377. J.Honerkamp, J.Weese, A non linear regularization method for the calculation of relaxation spectra, Rheol. Acta 32 (1993), 65-73. C.Carrot, J.Guillet, J.F.May, J.P.Puaux, Application of the MarquardtLevenberg procedure to the determination of discrete relaxation spectra, Makromol. Chem., Theory Simul. 1 (1992), 215-231. A.S.Lodge, A network theory of flow birefringence and stress in concentrated polymer solutions, Trans Faraday Soc. 52 (1956), 120-130. A.S.Lodge, "Elastic Liquids", Academic Press, 1964. B.Bernstein, E.A.Kearsley, L.J.Zapas, A study of stress relaxation with finite strain, Trans Soc. Rheol. 7 (1963), 391-410. B.Bernstein, E.A.Kearsley, L.J.Zapas, Thermodynamics of perfect elastic fluids, J.Research Nat. Bur. Stand., B: Mathematics and mathematical physics 68B (1964), 103-113. B.Bernstein, E.A.Kearsley, L.J.Zapas, Elastic stress strain relations in perfect elastic fluids, Trans Soc Rheol. 9 (1965), 27-39. A.Kayes, An equation of state for non-newtonian fluids, Brit. J. Appl. Phys. 17 (1966), 803-806. H.M.Laun, Description of the non-linear shear behaviour of a low density polyethylene melt by means of an experimentally determined strain dependent memory function, Rheol. Acta 1_/7(1978), 1-15. H.M.Laun, Prediction of elastic strains in polymer melts in shear and elongation, J. Rheol. 30 (1986), 459-501. M.H.Wagner, Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt, Rheol. Acta ~ (1976), 136-142. R.G.Larson, Convection and diffusion of polymer network strands, J.of Non-Newt. Fluid Mech. 13 (1983), 279-308. R.G.Larson, K.Monroe, The BKZ as an alternative to the Wagner model for fitting shear and elongational behavior of a LDPE melt, Rheol. Acta 23 (1984), 10-13.

196 29. M.H.Wagner, Elongational behaviour of polymer melts in constant elongation rate,constant tensile stress, and constant tensile force experiments, Rheol. Acta 18 (1979), 681-692. 30. M.H.Wagner, J.Meissner, Network disentanglement and time dependent flow behaviour of polymer melts, Makromol. Chem. 181 (1980), 1533-1550. 31. A.C.Papanastasiou, L.E.Scriven, C.W.Macosko, An integral constitutive equation for mixed flows: viscoelastic characterization, J. Rheol. 2_.7.(1983), 387-410. 32. M.H.Wagner, A.Demarmels, A constitutive analysis of extensional flows of polyisobutylene, J. Rheol. 34 (1990), 943-958. 33. M.H.Wagner, The nonlinear strain measure of polyisobutylene melt in general biaxial flow and its comparison to Doi-Edwards model, Rheol_Acta. 29 (1990), 594-603. 40 P.I~Currie, Constitutive equations for polymer melts predicted by the DoiEdwards and Curtiss-Bird kinetic theory models, J. of Non-Newt. Fluid Mech. 11 (1982), 53-68. 35. R.G.Larson ~Constitutive equations for polymer melts and solutions ~, Butterworths Publishers, 1988. 36. P.R.Soskey, H.H.Winter, Large step shear strain experiments with parallel disk rotational rheometers, J. Rheol. 28 (1984), 625-645. 37. M.H.Wagner, J.Schaeffer, Nonlinear strain measures for general biaxial extension of polymer melts, J. Rheol. 36 (1992), 1-26. 38. M.H.Wagner, J.Schaeffer, Constitutive equations from Gaussian slip-link network theories in polymer melt rheology, Rheol. Acta 31 (1992), 22-31. 39. M.H.Wagner, J.Schaeffer, Rubbers and polymer melts: Universal aspects of nonlinear stress-strain relations, J. Rheol. 37 (1993), 643-661. 40. G.Marucci, B.de Cindio, The stress relaxation of molten PMMA at large deformations and its theoretical interpretation, Rheol. Acta 19 (1980), 6875. 41. M.H.Wagner, S.E.Stephenson, The irreversibility assumption of network disentanglement in flowing polymer melts and its effect on elastic recoil predictions, J. Rheol. 23 (1979), 489-504. 42. M.H.Wagner, S.E.Stephenson, The spike strain test for polymeric liquids and its relevance for irreversible destruction of network connectivity by deformation, Rheol. Acta 18 (1979), 463-468.

197 43. R.G.Larson, Convected derivatives for differential constitutive equations, J. of Non-Newt. Fluid Mech. 24 (1987), 331-342. 44. N.Phan Thien, R.I.Tanner, A new constitutive equation derived from network theory, J. of Non-Newt. Fluid Mech. 2_(1977), 353-365. 45. N.Phan Thien, A non-linear network viscoelastic model, J. Rheol. (1978), 259-283. 46. R.I.Tanner, Some useful constitutive models with a kinematic slip hypothesis, J. of Non-Newt. Fluid Mech. 5 (1979), 103-112. 47. R.J.Gordon, W.R.Schowalter, Anisotropic fluid theory: a different approach to the dumbbell theory of dilute polymer solutions, Trans Soc. Rheol. 16 (1972), 79-97. 48. M.W.Johnson, D.Segalman, A model for viscoelastic fluid behavior which allows non-affine deformation, J. of Non-Newt. Fluid Mech. 2 (1977), 255270. 49. W.P.Cox, E.H.Merz, Correlation of dynamic and steady flow viscosities, J. Polym.Sci. 28 (1958), 619-622. 50. F.N.Cogswell, Converging flow of polymer melt in extrusion dies, Polym. Eng. Sci., 12 (1972), 64-73. 51. F.N.Cogswell, Measuring the extensional rheology of polymer melts, Trans. Soc. Rheol. 16 (1972), 383-403. 52. M.H.Wagner, A constitutive analysis of uniaxial elongational flow data of a low density polyethylene melt, J. of Non-Newt. Fluid Mech. 4 (1978), 3955. 53. R.Fulchiron, V.Verney, G.Marin, Determination of the elongational behavior of polypropylene melts from transient shear experiments using Wagner's model, J. Non-Newt. Fluid Mech. 4.~ (1993), 49-61. 54. M.Takahashi, T.Isaki, T.Takigawa, T.Masuda, Measurement of biaxial and uniaxial extensional flow of polymer melts at constant strain rates, J. Rheol. 37 (1993), 827-846. 55. Y.Einaga, K.Osaki, M.Kurata, Stress relaxation of polymer solutions under large strain, Polym.J. 2 (1971), 550-552. 56. D.C.Venerus, C.M.Vrentas, J.S.Vrentas, Step strain deformations for viscoelastic fluids: experiment, J. Rheol. 34 (1990), 657-682. 57. A.J.Giacomin, R.S.Jeyaseelan, T.Samurkas, J.M.Dealy, Validity of separable BKZ model for large amplitude oscillatory shear, J. Rheol. 37 (1993), 811-826.

198

58. A. Arsac, C. Carrot, J.Guillet, P.Revenu, Problems originating from the use of the Gordon-Schowalter derivative in the Johnson Segalman and related models in various shear flow situations,J. Non-Newt. Fluid Mech. 55 (1994), 21-36. 59. A.S.Lodge, J.Meissner, On the use of instantaneous strains, superposed on shear and elongational flows of polymeric liquids, to test Gaussian network hypothesis and to estimate the segment concentration and its variation during flow, Rheol. Acta 11 (1972), 351-352. 00 C.J.S.Petrie, Measures of deformation and convected derivatives, J. of Non-Newt.Fluid Mech. ~ (1979), 147-176. 61. R.G.Larson, A critical comparison of constitutive equations for polymer melts, J. of Non-Newt. Fluid Mech. 23 (1987), 249-269. 62. R.G.Larson, Derivation of strain measures from strand convection models for polymer melts, J. Non-Newt. Fluid Mech. 17 (1985), 91-110. 63. S.A.Khan, R.G.Larson, Comparison of simple constitutive equations for polymer melts in shear and biaxial and uniaxial extensions, J. Rheol. 31 (1987), 207-234. 4. R.I.Tanner,'Engineering Rheology ~, Clarendon Press, 1985.


199

Mathematical Analysis of Differential Models for Viscoelastic Fluids J. Baranger a, C. Guillop6 b. and J.-C. Saut c aLaboratoire d'Analyse Numfrique, Universit6 Claude Bernard and CNRS, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France bMathfmatiques, UFR de Sciences et Technologie, Universit6 Paris X I I - Val de Marne, 61 avenue du Gfnfral de Gaulle, 94010 Crfteil Cedex, France CAnalyse Num~rique et Equations aux D~rivfes Partielles, Universit6 Paris-Sud and CNRS, Bs 425, 91405 Orsay Cedex, France

1. I N T R O D U C T I O N .

THE MODELS

The complexity of viscoelastic flows requires a multidisciplinary approach including modelling, computational and mathematical aspects. In this chapter we will restrict ourselves to the latter and briefly review the state of the art on the most basic mathematical questions that can be raised on differential models of viscoelastic fluids. We want to emphasize the intimate connections that exist between the theoretical issues discussed here and the modelling of complex polymer flows (see Part III) and their numerical simulations (see Chapter II.3). As a matter of fact we do think that a better understanding of the mathematical properties of the models for viscoelastic fluid flows is fundamental in order to select a "good" constitutive equation, and to develop and implement numerical codes of practical use.

This chapter will be organized as follows. After a brief review of the classical differential models we will emphasize two important features of "Maxwell type models" (i.e., models without Newtonian viscosity), namely the instability to short waves and a "transonic" change of type in steady flows. Then we review the existence results of solutions known for steady flows and for unsteady flows. Afterwards we discuss the important topic of stability of flows. This issue is still wide open, in particular because of the presence of memory effects for viscoelastic fluids, contrary to the case of Newtonian fluids. Finally we conclude by presenting a few numerical schemes appropriate for simulating viscoelastic fluid flows, and we give some error estimates related to these schemes. For simplicity and because they are widely used in the numerical simulations, we will restrict ourselves to the class of differential models. Actually they display (at least qu~li*And Analyse Num~rique et Equations aux D~riv~es Partielles, CNRS and Universit~ Paris-Sud.

200 tatively) most of the striking phenomena observed in viscoelastic flows. These models obey the constitutive equations n

r = r ~ + rP, 7-" = 2r/,D, rP = ~ r i ,

(1)

i=1

- ~aT-i

ri + Ai~

+ [3i(D, 7-i) = 2r/iD , 1 < i < n,

where 7" is the (symmetric) extra-stress tensor, v is the velocity field, and D = D[v] = 1 ( V v + V v T) is the rate of deformation tensor. The coefficients r/j(> 0), 1 < i < n, and 2

--

ris(> 0) are viscosities; A/(> 0), 1 < i < n, are relaxation times. The tensor 7"s corresponds to a Newton/an contribution or to a fast relaxation mode. It plays a fundamental r61e in the mathematical and the numerical analysis; models with r/~ = 0 will be called "of Maxwell type", those with r/, > 0 will be called "of Jeffreys type". In the 7-i equation in (1) the symbol ~ - represents an objective derivative of a symmetric tensor, and is given by 9~r 0 = ( ~ . + ( v - V ) ) 7 " + r W - WT- - a ( D T + 7-D), Dt

(2)

1 where a, - 1 < a < 1, is a given real parameter, and W = ~(XTv - V v T) is the vorticity tensor. Finally ~ i ( D , vi) is a tensor-valued smooth function, which is at least quadratic in its two arguments in a neighborhood of 7- = 0, and submitted to restrictions due to objectivity. Most differential models of viscoelastic fluids reduce to (1), with an appropriate choice of the functions/3 i. Here are some examples, where for simplicity we only consider the case of o n e relaxation time, i.e. n = 1,/~11 - / 3 , )~1 ~- )~, and rh -- r/p. 1. 13 _= 0 corresponds to a version of Oldroyd models. W h e n in addition r/~ = 0, the particular values a = - 1 , 0, and 1 correspond respectively to the lower-convected, corotational (or Jaumann), and upper-convected Maxwell models. 2. ~ ( D , 7-) = 2Tr (7-D)a(Tr T)(T + I) and a = 1: Larson's model. Here a(TrT-) is a scalar function of Tr 7-. 3. /3(D,T) = a rTr(7-), with a constant: this is a version of the Phan-Thien and Tanner model. Larson generalized it to /3(D, v) = a7 "2 + ~7-, where a and ~ are scalar functions of Tr 7" and v. Note that the most general version of the PhanThien and Tanner model is when /3(D, 7 - ) = v ( - 1 + e x p ( a T r 7-)) with a positive constant. 4. fl(D, v) = a 7"2, cr constant" Giesekus' model. It is a particular case of 3. 5. ~ ( D , r ) = ~o(Tr r ) D - p l ( r D + D r ) + vlTr ( r D ) I + #2D 2 + v2Tr (D2)I, where #0, #1, Vl, #2, v2 are some constants: S-constant Oldroyd model.

201 6. White--Metzner's models correspond to/3 = O, Aand % being some given functions n--1

of the second invariant II = I I D = ~Wr (D2). Typically, r/p = 70(1 + (AoII)2"} "~'~, n > 0, which corresponds to Carreau's law. Of course this list is not exhaustive. (See other models in [1,2].) Also models with internal variables (order parameters) as those of [31 can be put in a similar (though more complicated) framework. In particular, there are additional constitutive equations of differential type for the order parameters [4]. Equations (1) are to be solved together with the equations of conservation of momentum and mass: 0v

+ (v. v),,) + vp

= div r + f,

(3)

div v = 0, and appropriate initial and boundary conditions. Compressibility effects cannot always be neglected for polymer flows (see e.g. [5]), equations (3) could be replaced by the corresponding equations for slightly compressible fluids. (See section 3.3 below.) Solving the previous set of equations, especially with realistic boundary conditions, is a formidable task and a lot of issues are still unanswered. This is not surprising because of the complexity of the equations, and because of their recent derivation, around 1950 for the first nonlinear models, the Oldroyd models. On the other hand, the mathematical theory for the Euler and the Navier-Stokes equations for incompressible Newtonian fluids is still not complete though these equations were derived in 1755 and 1821 respectively!

2. M A X W E L L

T Y P E M O D E L S : L O S S OF E V O L U T I O N

AND CHANGE

OF T Y P E Maxwell type models (r/$ = 0) display two striking phenomena which are not present in Jeffreys type ones (r/$ > 0), and which will be described now. 2.1. Loss of evolution: H a d a m a r d instabilities or instabilities to s h o r t waves This section concerns the Cauchy problem or initial value problem, where initial data at time t = 0 are given. It was noticed by Rutkevitch [6,7], and systematized by Joseph et el. [8], Joseph and Saut [9], and Dupret and Marchal [10] that Maxwell type models can present Hadamard instabilities, that is, instabilities to short waves. (See [11] for a recent discussion of more general models.) Then, the Cauchy problem is not well-posed in any good class but analytic. Highly oscillatory initial data will grow exponentially in space at any prescribed time. An ill-posed problem leads to catastrophic instabilities in numerical simulations. For example, even if one initiates the solution in a "stable" region, one could get arbitrarily close to an "unstable" one. For instance, the Maxwell model

202

"~a "/"

A---~-- + r = 2r/D,

p( v

+

v > ) = div

(4)

v,,

div v - 0, is ill-posed whenever 1 - a h m ~ , - +_.._.~a 1 A~t, > .~, (5) 2 2 where hm~ (resp. hmtn) is the largest (resp. smallest) eigenvalue of r. Inequality (5) is actually equivalent to the fact that the (second order in t and x) equation for the vorticity is ill-posed. It can be shown [8] that Hadamard instabilities are possible for admissible motions if a is in the interval ( - 1 , 1 ) , e.g., in extensional flows. On the other hand, restrictions on the eigenvalues of r prevent Hadamard instabilities for a = +1. This is immediately seen from the integral forms of (4)-(5) for the upper- and lower-convected Maxwen models, which imply constraints on the eigenvalues of the Cauchy-Green tensors. (See, for instance,

[121.) 2.2. Change fluids

o f t y p e in s t e a d y flows" a t r a n s o n i c p h e n o m e n o n

in v i s c o e l a s t i c

The partial differential equations system for steady flows of Maxwell type (i.e., with o _ 0) is of composite type, neither elhptic, nor hyperbolic. This is not surprising, the same being true for instance for the stationary system of ideal incompressible fluids. The new feature, discovered in [8], is that some change of type may occur. In fact an easy but tedious calculation shows that three types of characteristics are present: - complex characteristics (elhptic part) associated with incompressibility; - real characteristics (hyperbolic part) associated with the propagation of information along streamlines (double in 213, of multiplicity 4 in 3D); - characteristics which change type: complez if and only if the equation for the vorticity is elliptic, real if it is hyperbolic. The change of type (analogous to the well known situation in gas dynamics) occurs t - - - -

when the modulus Iv(x)l of the velocity exceeds ~/-~, which is the speed of propagation v r--

shear waves in the fluid at rest. In other words, if one introduces a viscoelastic Mach number M = Re We (see Section 3 below for the definitions of Re and We), the flow goes from sub- to supercritical as M crosses 1. Then the vorticity equation changes from elliptic to hyperbolic, and there are waves of vorticity. This, of course, implies a qualitative change in the nature of the flow, which is supported by some experiments (e.g., [13]). It is interesting to note in this context that the experiments of Metzner et hi. [14] can be interpreted as suggesting a discontinuity in the vorticity. This change of type leads to drastic changes in the boundary conditions. (See Section 3.) Very few (mathematical or numerical) results are known in the supercritical case. (See [15-17].) Among models without Newtonian contribution, Maxwell-like models possess the nice feature that the change of type is associated with a change of type in the vorticity. A of

203 general class of differential models sharing this property was exhibited in [8]. This paper (and [9,18])also contains a classification of classical flows (Couette, Poiseuille, extensional, ...) for various Maxwell models with regard to type. To illustrate the generality of change of type in elastic fluids (i.e., fluids whithout Newtonian contribution) we present here a brief analysis of the linearization around a uniform flow v = (U, 0, 0), using the ColemanNoll theory of fading memory [19]. Stemming from the rather general concept of a simple fluid (Noll), this theory provides a systematic way to derive constitutive laws for special flows (e.g., perturbations of rigid motions). The idea is to choose a ]L2-weighted space for the history of deformations (see [20] for other choices of function spaces), and to use the Riesz theorem and isotropy to express the derivative of the stress at a given motion. One gets in this fashion constitutive laws of the type r =

~0 ~

x(s)[Ct(t

-

s)

-

I]ds,

where x is a scalar kernel (the relaxation kernel) satisfying ~/h 2 ~. L2(O, cx~), Ct is the right relative Cauchy-Green tensor, and h is the weight function associated to the space of histories of deformations. For example, the upper-convected Maxwell model, where a = 1, corresponds to x(s) - 77 exp(-s/A). It can then be shown that the vorticity ~ of the linearized flow around the uniform flow v = (U, 0, 0) satisfies the equation

02'~2 - x(0)A~ = lower order terms. pU2 b-~z

(6)

02 02 Define the differential operator in the plane perpendicular to the flow by A• = ~ + az----~, the Mach number M = U/c, and the speed of shear waves c = ~x(0)/p. Then equation (6) reads 02( - A• (M 2 - 1) ~'x2

- lower order terms,

which shows that the vorticity changes type from elliptic to hyperbolic when M crosses 1. For the upper-convected Maxwell model, the full equations for ( reads

02~ (M 2 - 1)~-7x2 - A •

+

M 0~

cA Ox

--0~

with c : ~/~o" (See [21].) u i A change of type analysis for more complicated models is performed in [22]. 3. S T E A D Y F L O W S First equations (1)-(3) are written in a nondimensional form (see [23] for the details),

204

Ov Re (-~- + ( v - V ) v ) - (1 - e)Av + Vp = d i v 1" + f, Or We(-~- + (v-V)1" + fl(Vv, r ) ) + v = 2eD,

(7)

d i v v = 0. The parameters in these equations are the Reynolds number Re = pUL/r/(U and L are a typical velocity and a typical length of the flow, and 7/= r/s + r/p is the total viscosity of the liquid), the Weissenberg number We = AU/L, and the retardation parameter e = r/p/r/. Obviously, 0 < e < 1; e = 1 corresponds to Maxwell-type fluids, and 0 < e < 1 corresponds to Jeffreys-type fluids. Observe the change of notation in equation (7), w h e r e / 3 ( V v , r ) denotes now all the nonlinear terms in Vv and r other than the term ( v . V ) r . f denotes some given body forces. To start with, we consider steady flows of Maxwell-type fluids in a bounded smooth domain 9t of ]RN, N = 2, 3, and with simple boundary conditions, namely the system Re (v- V ) v + Vp = div r + f, div v = 0, (s)

W e ( ( v - V)7" + ~ ( V v , 1")) + r = 2D, Vl0a = v0, v 9nloa = 0, where n denotes the outward unit normal vector to the boundary Oft of ~, and where v0 is a prescribed velocity field satisfying div v0 = 0 in 9t and v0 9nl0a = 0. The following result is due to Renardy [24]. T h e o r e m 3.1 ( E x i s t e n c e of slow flows) Let Ilfll~ a~d b~ sufficiently small. Then there exist v E I-I3(a), 1" E l-I2(f~), and p E H2(f~) solution of system (8), unique among all small solutions. If moreover f E I-Ik(ft), Vo E l-Ik+X/2(Of~), for some integer number k > 2, then v E

Ilvoll.~/~(oa)

,- e

p e

Above and throughout this chapter [[-ilk, for k integer, will stand for the norm in the Sobolev space Hk(9/) (space of real functions defined in ~) or l-Ik(Ft) (space of vector or tensor valued functions defined in ~). The idea of the proof of Theorem 3.1 is to use an iterative scheme which alternates between a perturbed Stokes system (corresponding to the elliptic part of the system (8)) and a hyperbolic equation whose characteristics are the streamlines. This result has been extended in several ways. 3.1. W h i t e - - M e t z n e r m o d e l s Hakim [25] has shown that Theorem 3.1 is also true for White-Metzner models, where r = ~.s + rv satisfies 1"s = 2r/~D, r/~ _> 0 (constant), ~a q'p 7"P "4- "~II ~ --= 2r/IID,

205

1 provided that )~II and r/u are smooth functions of II = ~Tr (D 2) such that )~II > 0 and r/II > 0. 3.2. W e a k l y elastic fluids Let ~ be a steady solution of the Navier-Stokes equations (with prescribed body forces and zero boundary velocity). Note that ~ is not assumed to be "small". Then, there exists a steady solution (vc, re, pc) of any Jeffreys model with a sufficiently small Weissenberg number We, and with a sufficiently small retardation parameter e, such that (vc, re) is close to (~r O) and e close 0. (See [26].) 3.3. Slightly c o m p r e s s i b l e fluids In the case of sligthly compressible fluids, and with the hypotheses that the flow stays at low pressure and isothermal, system (7) is replaced by 0v Re ( ~ - + ( v . V)v) - (1 - e)(Av + Vdiv v) + Vp = div 7 + f,

+ (v.

+Iv

+

+

(Vv,

+

= 2 o,

(9)

aivv- 0,

where fl > 0 is a large parameter characterizing the (slight) compressibility. In the case of steady flows of slightly compressible fluids, R. Talhouk [27,28] considers a weak version of (9), where the conservation of mass equation is replaced by a conservative steady equation, rI((v, v ) p ) +

div,, = 0,

and II denotes the projection on zero mean value functions, defined by H(g) = g (fn9 dz)/lfll for a scalar function g. He shows that the corresponding system admits a unique steady regular solution (v~, r~,pz) in both cases, Maxwell (c = 1) and Jeffreys (0 < c < 1), provided that the parameter/3 is large enough. Moreover, for fixed ~, 0 < < 1, the slightly compressible solution converges to the incompressible one (satisfying the steady equations associated to (7)) when/3 goes to oo. The proof of the existence results is based on the study of a linearized system and on the application of the Schauder fixed point theorem. The convergence to the steady incompressible limit is obtained by proving that the solution is bounded independently of /3 large. (See [27,281.) 3.4. P r o b l e m s with inflow b o u n d a r i e s It is well known that, for the Navier-Stokes equations, the prescription of the velocity field or of the traction on the boundary leads to a well-posed problem. On the other hand, viscoelastic fluids have memory: the flow inside the domain depends on the deformations that the fluid has experienced before it entered the domain, and one needs to specify conditions at the inflow boundary. For integral models, an infinite number of such conditions are required. For differential models only a finite number of conditions are necessary (more and more as the number of relaxation times increases, ...). The number

206

of conditions depends on the model--Maxwell- or Jeffreys -type--and on the flow---subor supercritical. Determining the nature of the boundary conditions is by no way a trivial matter and no complete answer is known so far. We treat briefly two different approaches.

3 . 4 . 1 . P e r t u r b a t i o n s o f a u n i f o r m flow. The first example, due to Renardy [17,29], deals with a special nonlinear situation, namely that of a small perturbation of a uniform flow v = (U, 0, 0) transverse to a strip, as shown on Figure 1.

v=(U, O, O)

X

x=O

x=1

Figure 1" Uniform flow in the strip {0 < x < 1} The problem consists in finding the appropriate conditions to prescribe for the extrastress tensor at the inflow boundary {z = 0}, so that the steady problem is well-posed. The method of analysis is a variant of the algorithm leading to Theorem 3.1. For Jeffreys models, where r/$ > 0, all the components of the elastic part r p of the extra-stress can be prescribed at {z = 0}. For Maxwell models, where r/$ = 0, we need to distinguish the sub- and the supercritical cases. In the subcritical case (i.e., U < v/rl/(p,~)) and in two space dimensions, one can prescribe the diagonal components ~rP and 7 p, whereas in three space dimensions a correct choice of boundary conditions for rP is not simple. A possible choice of four boundary conditions is a nonlocal one (in terms of the Fourier components of rP--see [29]). An alternative approach leading to first order differential boundary conditions at the inflow boundary is described in [301. For Maxwell models in the supercritical case (i.e., U > x/rl/(P~)), the previous choice of boundary conditions leads to an ill-posed problem (as does the Dirichlet boundary condition for a hyperbolic equation), as shown in [17]. In addition to the normal velocities at both boundaries (inflow and outflow) and to the previous inflow conditions on the stresses, one can prescribe the vorticity and its normal derivative in two space dimensions, or the second and third components of the vorticity and their normal derivatives in three v

207

space dimensions. A discussion of the traction boundary conditions--where the total normal stress is prescribed on the inflow and outflow boundaries--for Jeffreys-type fluids is given in [31], and for Maxwell-type fluids in [32]. a.4.2. A b s o r b i n g b o u n d a r y conditions for viscoelastic fluids We briefly present here some results taken from Tajchman's doctoral thesis [33]. Interest is focused on models described by (see equations (7) and (9)) Re ( - ~ + ( v - V ) v ) - ( 1 - e ) A v + Vp = div r + f, Or We (-~- + (v. V ) r +/3(Vv, r)) + r = 2eD, d i v v = 0 , or

(10)

0p

-~+/3divv=0,

according to whether we consider the incompressible case (/3 = oc) or the slightly compressible case (/3 > 0). The geometry of the flow is supposed "infinite" (i.e. very large in one direction), such as in the case of a flow around an obstacle, or in a long die. The flow "at infinity" is assumed to be known (uniform, Poiseuille flow, ...). For computational purposes, one introduces artificial boundaries, at a finite, hopefully not too large, distance. The problem is to define which conditions to impose on the artificial boundaries in order to obtain a solution of the truncated problem, which is as close as possible to the solution of the original problem. Such considerations where first carried out by Engquist and Majda [34] for wave equations (linear hyperbolic equations), and developed by many authors later on. In particular Halpern [35] considered the case of parabolic perturbations of hyperbolic systems. From physical considerations (plane waves travelling through the fluid), one gets boundary conditions which make the artificial boundaries transparent to the waves leaving the computational domain and which absorb the waves entering the domain (other than those generated by the solution at infinity). We suppose that the artificial boundaries are far enough away in order to justify the linearizations around the uniform solution at infinity. The linearized problem reads 0v Re (-~- + ( v ~ . V)v + (v. V)v~) - ( 1 - e)Av + Vp = div r,

vo~

(11) Op

d i v v = 0 , or ~ - ~ + / 3 d i v v = 0 . Looking for plane wave solutions amounts to testing nontrivial solutions of the type N

U(s. J)exp[ir + ~(s, w/)xl + ~ ~bkxk)], k=2

where ~b = ( ~ 2 , ' " , ~bN), and where outwards pointing.

X1

is the normal direction to the artificial boundary,

208

The sign of the real part of these solutions determines the directions of propagation of the corresponding wave. For each wave entering the domain of computation one imposes a boundary condition which eliminates it: = 0,

where V is the corresponding left eigenvector, and ~ denotes the Laplace transform of v with respect to t and the Fourier transform with respect to x' = (x2,... ,XN). These conditions are not local (they are integral relations difficult to incorporate in a numerical code). By perturbation techniques one gets local approximations which are partial differential equations on the boundary. Higher order approximations can be obtained at the price of increasing difficulty in the computations. To be of any practical use, the artificial boundary conditions have to be stable: in particular, they should not depend on rounding errors. The stability analysis is made on the linearized problem by computing the time evolution of some solution norms. Finally one can compare the absorbing conditions which are obtained in different cases, e.g., the limit as e goes to 1 or as ~ goes to infinity. The results crucially depend on the subcritical or supercritical nature of the flow. We refer to [33] for a precise description. 3.5. T h e r e - e n t r a n t c o r n e r s i n g u l a r i t y So far only domains of the flow with smooth boundaries have been considered. However, re-entrant corners as in a sudden 4:1 contraction are well-known to give rise to numerical difficulties in the numerical simulation of viscoelastic flows. (See e.g., Chapter I1.3.) The analysis of the corner singularity is delicate. We refer to the recent works of Hinch [36] and Renardy [37,38], who have contructed a matched asymptotic expansion for the steady solution to a Maxwell fluid flow near the corner. 4. U N S T E A D Y

FLOWS

Existence results for unsteady flows are important in two ways. First, the (local) well-posedness of the initial and boundary value problem proves the adequacy of a given model to describe (at least locally) dynamical situations. Second, global well-posedness is preliminary to any nonlinear stability study. 4.1. F i x e d g e o m e t r y We first consider Jeffreys-type models, namely system (7) with e < 1, which is complemented with the Dirichlet boundary condition vl0~ = 0,

(12)

and initial values V [ t = 0 --" V0~ 7"It=0 - " 7"0-

Note that here v0 and 7"0 are not assumed to be "small". In what follows we shall denote the $obolev spaces of real functions, vector or tensor valued functions previously defined, by H k = Hk(~) or l-Ik = l-Ik(~t). To start with we state a local existence theorem [23].

209

T h e o r e m 4.1 (Local existence of u n s t e a d y flows) Let Ft be a bounded domain of R N, N = 2, 3 with C,3 boundary. Let f e L~oc(R+; tta), f' E L~oc(]R+; I-I-a), v0 E I-I2 R tt~, with div v0 = 0, and ro E I-I 2. Th~. t h ~ ~i~t T" > 0 ~.d ~ ..iqu~ ~ol.tio. (v. r.p) of ~u~t~m (~), (1~). ~.d (lS), which satisfies v e L2(0, T*; I-I3) nC([0, T*); I-I2 N IH~), v' e L2(0, T*; I-I~)N C([0, T*);L2), p e L2(0, T*; H2), r e C([0, T*); H2). A similar result has been proven by Hakim [39] for a class of White-Metzner models (still with e < 1) under suitable assumptions on the constitutive functions $(II) and y(II). These assumptions are satisfied in particular by the Carreau and the Gaidos-Darby laws [40]. R e m a r k 4.1 The results of Theorem 4.1 do not depend on the precise nature of the term fl(Vv, I") and thus are model independent. In particular, these results are still valid for differential models with internal variables [4]. R e m a r k 4.2 If more regularity is assumed on the data, then more regularity of the solution is obtained provided an additional compatibility condition on the initial values is imposed. We now turn to local existence of solutions for Maxwell-type models. The situation is much trickier here since these models can display Hadamard instabilities (see Section 2.1), and no general results seem to be known so far. One has, in any case, to restrict initial data to "Hadamard stable" ones. A possible way to overcome the difficulty is to consider models satisfying an ellipticity condition, which will imply well-posedness. This approach was followed by Renardy [41], whose results are briefly described below. The extra-stress tensor 7" = (rij) is assumed to satisfy an equation of the form + v.

=

Ovk

+

where (due to frame indifference) 1 A o k t ( v ) = -~(,Sikr,3 - ~Silrk3 -- rik~Sl3 + ri,6kj) + B,jkt(~'); the tensor (Bijkt) is symmetric in k and l (and of course in i and j), and satisfies Bi3kz = Bklij. We set 1

C~jkz = Bokl + ~(*ikrU -- *~rkj -- r~k&j -- T~t6kj), so that Aip:t = 7it6kj + Cijm, and Cijkl "-- Cklij. The crucial hypothesis is the strong ellipticity condition,

o~,(~-)r162

>_ ,~(~-)1 0. Note that, for Maxwell models with - 1 _< a < 1, relation (15) is satisfied locally in time provided it is satisfied at time t = 0. For a = + l , relation (15) is equivalent to relation (5), which insures that the initial value problem is well-posed: this is a natural condition to impose on the stress. But, for a 5r + l , condition (15) reveals that the model is not always of evolution type, which means that Hadamard instabilities can occur. (See Section 2.1) Under hypothesis (15) and under some smothness conditions on the (bounded) domain of the flow, and on the functions Aijkl and gij, Renardy [41] proves the local existence and the uniqueness of a I-I~ fl ][-I4 solution of equations (3) and (14), provided that the initial data v0 and r0 are smooth, and satisfy a compatibility condition at time t = 0. In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. Another local in time existence result concerns the case where the domain of the flow is unbounded, but satisfies certain uniform regularity conditions [27]. This result extends the result of Theorem 4.1, but in a context where there is non-compactness. Talhouk [27,43] also has studied the local existence of flows in a bounded two dimensional channel f~ = (0,1) x (0, L), for which the inflow v_ and the outflow v+ are given and satisfy the following: there exists an a > 0 such that v_ - n < - a on F_ = Off_ , and v+ 9n > 0 on F+ = 0fl+. (See Figure 2.)

v+

V

r

F

F+

0

1

x

Figure 2: The domain D, = (0, 1 ) • (0, L) with inflow and outflow boundaries, I'_ and F+

The flow is assumed to be periodic in the y- and the z-directions. Moreover the extrastress components are given on the inflow boundary. Assuming that the data on the inflow boundary satis~- a compatibility condition. Talhouk shows the existence of a local in time solution in the bounded channel.

211

Concerning global (in time) existence of solutions, the first general result is the following [23], which is established for Jeffreys models with a sufficiently large Newtonian contribution to the extra-stress. T h e o r e m 4.2 (Global e x i s t e n c e of flows w i t h small d a t a ) Let fl be a bounded domain in 1~N, N = 2,3, with a C 4 boundary. There exists a parameter eo depending on f~, 0 < eo < 1, such that if 0 < e < eo and if v0 E I-I2N I-I~, with div v0 = 0, f E L~ I-IZ), f' E L~(R+; I-1-1) are small enough, then system (7), (12), and (13) admits a unique solution

v e c~(R+; H ~ n ~ ) n L~or v' e C~(l%; L ~) n L~or

~), ~I~),

The proof of this result is based on an energy method: one shows that the quantity

Y(t)

= (1 - e ) 2

-----~~e 2 Jlv(t)ll~ + Re IJv'(t)flo2 +

We(1 - e)3 We f]r(t)r[~ +--][r'(t)JJ~ s Re ae3

satisfies the inequality Y' + a o Y O, a l > O,

where a2 > 0 is small when the data are small. The conclusion is reached when noticing that a function Y which satisfies the above differential inequality and has small initial value Y(0) stays bounded for all times. A similar result for White-Metzner models is proven in [25] under the same hypotheses on the constitutive functions as for the local existence result mentioned earlier, plus the hypothesis A(z) _< M, for all x in IR+. R e m a r k 4.3 The restriction 0 < e < e0 is due to the treatment of the boundary conditions in the linear coupled terms. R e m a r k 4.4 No result such as Theorem 4.2 seems to be known for Maxwell models. We however have to mention the result [44], where the upper-convected Maxwell model in the whole space IR3 is considered. In the context of Theorem 4.2, one can prove that two solutions, which are sufficiently small at t = 0 and correspond to sufficiently small body forces, will get asymptotically exponentially close as t goes to oc. Using this stability result and classical arguments ([4.5,46]) one obtains the following result. C o r o l l a r y 4.1 ( E x i s t e n c e of small stable T - p e r i o d i c or s t e a d y flows) Let ~ and f~ be as in Theorem 4.2.

212

(i) Let f E L~176 ]H1), f' E L~176 ]H-1) be time periodic of period T > O, and small enough. Then there exists a T-periodic solution of system (7), (12), unique among the small T-periodic solutions. (ii) If moreover f is time independent, there exists a steady solution, which is unique among small steady solutions. (ii O The aforementioned solutions are non-linearly stable. (See also Section 5.) R e m a r k 4.5 Corollary 4.1 obviously implies the solution) for e not to close to 1. Using a different removed this restriction and showed the Liapunov 0 < e < 1. This fact is not known for Maxwell-type

stability of the rest state (the zero approach, Renardy [42] has recently stability of the rest state for all e's, models (e = 1).

R e m a r k 4.6 A number of problems are still open. For example the existence of (small) periodic solutions for Maxwell models is unknown, as is that of arbitrary (not small) periodic solutions for Jeffreys models. For example too, nothing is known concerning the global existence of unsteady solutions for differential models in two or three space dimensions (say, weak solutions, singularities in finite time, . ..). See below, for some examples in one space dimension. More specific results can be obtained in some one dimensional situations, which we describe now. Following [47], we consider shearing motions of an Oldroyd fluid, such as Couette or Poiseuille flows. The dimensionless equations are easily reduced to a system for the shear component of the velocity v(x,t), the shear stress r(x,t), and a linear combination of normal stresses a(x, t), x E I, t >_O, Re vt - (1 - e)v** = r . - f,

+

T

=

s

For Couette flow, one has I = (0,1), and f = 0, while for Poiseuille flow, I = ( - 1 , 1 ) , and f = 1 is the constant pressure gradient in the flow direction. The boundary conditions are v ( - 1 , t ) = v(1,t) = 0, t _> 0, for Poiseuille flows, or v(O,t) = 0, v(1,t) = 1, t _> 0, for Couette flows.

(17)

The parameter a will be assumed to satisD - 1 < a < 1. The crucial observation is the following a priori bound on the stress components: -

+ (1 - aZ)v2(x, t)

Using the bound (18), we can then prove the following result.

(is)

213

Theorem 4.3 (Existence and uniqueness of one-dimensional global flows) Let 0 < e < 1. (i) Uniqueness. There exists at most one solution (v, a, T) of system (16)-(17) in the H1)] x [L~ x ]R+)]2. space [L~(]R+; L 2) N L~or (ii) Existence. Let v0, a0, TO E H ' (I) such that Vo satisfies the boundary conditions (17). Then for any T > O, there exists a unique solution ( v , a , r ) of system (16)-(17) in the space [C([0, T]; H ~) N L2([O,T];H2)] x [C([0, T); H~)] 2. Moreover v e Cb(R+; L 2) and the bound (18) holds true.

Another proof of this result is obtained by Malkus et al. [48]. R e m a r k 4.7 In the case e = 1 (Maxwell models), system (16)-(17) is not always of evolution type. (See section 2.1.) Indeed, Renardy et al. [49] have constructed initial data in the hyperbolic domain, with steep gradients, such that the velocity and the stress develop singularities in their first space derivatives in finite time. The idea is to reduce the system, by a clever change of variables, to a degenerate system of three nonlinear hyperbolic equations. R e m a r k 4.8 The results of Theorem 4.3 depend crucially on the model (the Oldroyd model). It would be interesting to know what happens for one dimensional flows of general differential models with a Newtonian contribution. Similar results can be obtained for Couette or Poiseuille flows of several fluids in parallel layers: these flows are important in particular in the modelling of coextrusion experiments. Le Meur [50] has studied the existence, uniqueness and nonlinear stability with respect to one dimensional perturbations of such flows. The behaviour of each fluid is governed by an Oldroyd model such as (16)-(17), where the nondimensional numbers Re and We are defined locally in each fluid. On the rigid top or bottom walls, the velocity is givenPzero on both walls for Poiseuille flow, and zero or one depending on the wall for Couette flow. The interface conditions on the given interfaces are ~7"aim(-pI + 2(1 - e)D + r I = 0, where [-] denotes the jump of a quantity across the interface, and "/'dim ~U/L is defined in each fluid. One can also show that all one dimensional time-dependent perturbations of a steady multifluid flow exist for all times, and stay boundedmas in the case of one fluid. Similar results can be obtained for axisymmetric Poiseuille flows of several fluids. A similar study is also made for plane Poiseuille or Couette flows of several fluids having a Phan-ThienTanner constitutive equation [50]. =

4.2. Free b o u n d a r y p r o b l e m s In many practical situations the geometry of the domain occupied by the fluid is not given. In particular, there are free boundaries such that interfaces between air and liquid, interfaces between several liquids,...

214

Mathematical results for these flows are not simple, even for Newtonian fluids. In [50,51], Le Meur has considered unsteady flows of an incompressible fluid of Jeffreys-type submitted to surface tension above a fixed rigid bottom. He proves that there exists a unique local solution with a free surface, which is close to a flow with a given flat interface. 5. S T A B I L I T Y I S S U E S Instabilities are one of the main challenging problems in the mathematical theory of viscoelastic fluids. We shall only consider bounded geometries. Let us briefly review the goal and the main difficulties of this subject. One wants to explain (predict, avoid, ...) the instabilities occurring in polymer processing. When polymer melts are extruded from a pipe, instabilities often occur at a critical value of the wall shear stress. They are known as spurt flows, shark skin defects, melt fracture, ... They manifest themselves in a jump of the flow rate for a given pressure gradient, irregularities on the surface of the extrudate, pressure oscillations, chaotic behaviour, ... (See [52-54], and Chapter III.4.) Their physical explanation is not well understood. Possible causes could be slip at the wall (interaction of the fluid with the wall of the pipe), or propagation of defects in the pipe, ... Mathematical explanations could be change of type, or constitutive instabilities (non-monotone shear stress / shear rate curve), ... In any case viscoelastic instabilities differ from the classical "hydrodynamic instabilities" which occur in Newtonian flows at high Reynolds numbers. Those instabilities happen at moderate Reynolds numbers, and are basically due to the elastic effects. (See a very good review in [55].) Note that the viscoelastic analogue of the classical hydrodynamic instabilities (B6nard and Taylor experiments) can in principle be studied, at least formally, by the methods of bifurcation theory. (See the recent work of Renardy and Renardy [56] on the B~nard problem, and of Avgousti and Beris [57] on the Taylor-Couette problem.) From a fundamental point of view, none of the theoretical results necessary to justify rigorously bifurcation or stability studies have been fully established so far for the equations governing viscoelastic flows (contrary to the Newtonian case). Such results concern, for instance, the relations between linear stability and the spectrum of the associated operator, between linear and nonlinear stability, or the reduction of the dynamics to a centre manifold. In order to explain and clarify the mathematical issues we recall some classical facts. 5.1. G e n e r a l i t i e s The equations for a perturbation u of a steady solution u5 of an incompressible viscoelastic fluid can be written as an abstract equation in a Hilbert space X, = A~ + f(~),

(19)

u ( o ) = Uo,

where u C X is a vector whose components are the velocity and extra-stress components. (The pressure can be eliminated by projection onto a space of divergence free vectors.) A is the linearized operator around u~. f(u) the nonlinear part (at least quadratic in u), and u0 the perturbation at time t = 0. The boundary conditions are taken into account

215 in the space X. The problem of the stability of the steady flow u, is expressed by the three following notions. 1. ($1) Nonlinear stability (or Liapunov stability). For any 51 > 0, there exists 52 > 0 such that for every u0 with Ii~011x < ~=, the corresponding solution u(t) of (19) exists for all t > 0, and satisfies II~(t)llx _< 5x, for ~11 t > 0. If furthermore Ilu(t)llx goes to 0 as t goes to ~ , u, is said asymptotically stable. 2. ($2) Linear stability. For any v0 in X, the solution v(t) of the linear problem

~t =Av,

(20)

v ( o ) = vo,

satisfies lim IIv(t)llx = o. 3. ($3) Spectral stability. The spectrum a(A) (i.e., the set of complex numbers ~ such that the operator )~1 - A: D(A) ~ X is not invertible, where D(A) is the domain of A) is contained in the left half plane { ~ < 0 }. The strongest notion (and the most "physical" one) is (Sl). On the other hand (S3) is the easiest to check: it "suffices" to compute, numerically in general, the spectrum of the linear operator A. This is essentially the classical Orr-Sommerfeld approach. (See Section 5.5 for an example.) Unfortunately, in general, there is no relation between these three notions. The link between ($2) and ($3) is in relation to a formula of the type

a(S(t)) = exp(ta(A)), t >_ O,

(21)

where S(t) is the semi-group generated by A (i.e., the family of operators in X, which associate v0 to the solution v(t) of equation (20)). If relation (21) holds true, then it is clear that (S3) implies ($2). In general, relation (21) is false: one only has exp(tcr(A)) C e(S(t)). The corresponding formula is true for the point spectrum (constituted of eigenvalues), and for the residual spectrum, but it is false for the continuous spectrum: there exist abstract operators A with empty spectrum, but such that the continuous spectrum of S(t) is the circle of radius exp(Trt); thus the solutions of equation (20) grow exponentially in time. Other than for the classical case of the finite dimension, it is known that relation (21) is true for compact or analytic semi-groups. This covers the case of Newtonian flows, where the associated system of partial differential equations is parabolic. In this case, one also has the implication "($3) ==~ (S1)". (See [58] for instance.) The main question concerning the implication "($3) ~ ($2)" is to know whether or not the pathological situations of the aforementioned abstract examples can occur in "concrete" situations, in particular in those occurring in the study of linear stability for viscoelastic flows. In fact, because of memory effects, the underlyingsystem is (at least)

216 partially hyperbolic, and we cannot hope that S(t) be compact or analytic (except for the one dimensional flows described in Section 5.3, where the hyperbolic part degenerates.) A good situation to first at look is the one of linear hyperbolic equations or systems. Actually the implication "($3) ~ ($2)" has been proven in [59] for certain hyperbolic systems in one space variable. A more general (and simpler) proof has been given by Renardy [60]. On the other hand, Renardy [611 has constructed a simple example (namely the wave equation utt = u** + u,~ + eiYu,, with periodic boundary conditions), showing that the linear stability of hyperbolic partial differential equations in two or more space dimension is not necessarily determined by the spectrum of the operator. We now turn to the case of viscoelastic fluids. 5.2. Linear stability There are very few results concerning the linear stability (with respect to two or three dimensional perturbations) for viscoelastic flows, essentially by lack of a general result "(S3) ~ (S2)'. In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the assumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (i.e., condition ($3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, and does not generalize so far to other Maxwell-type models. In another paper M. Renardy [64] considers the plane Couette flow of an Oldroyd-B fluid (i.e. a Jeffreys model associated to the upper-convected derivative). He proves linear stability under the (theoretical) assumption that the eigenvalues have negative real parts, condition ($3) implying condition ($2) in this context. Note that the Couette flow is always stable in the Newtonian case [65], and numerical calculations (on the OrrSommerfeld side) suggest that this may be so for the Maxwell and Oldroyd-B models as well [66]. Again the structure of the model and of the flow is made use of in a crucial way (in fact one uses a factorization of a differential operator which was discovered by Gorodtsov and Leonov [63]). This result of stability has been extended by Renardy [61] to any parallel shear flow with a strictly monotone profile--thus excluding Poiseuille flows-for an arbitrary fluid of Jeffreys-type. Finally we comment briefly on weakly elastic fluids. (See [26] and section 3.2.) We assume that the given Newtonian solution ~r satisfies ]1~r < (ClRe)-X, where Cl is some constant depending only on the domain of the flow: this condition ensures that ~ is asymptotically Liapunov stable. (See e.g., [67].) Then, the viscoelastic solution (v,, ~',) close to (~, 0) is linearly (asymptotically) stable for e > 0 small enough. One dimensional flows It turns out that one dimensional perturbations to a viscometric flow lead to a system for which the implication "($3) ~ ($2)" is true. For plane laminar flows, such as Couette or Poiseuille flows (see Figure 3), the velocity and the stress depend only on the first coordinate x E I, where I is the interval (0,1) for 5.3.

217

Couette, and ( - 1 , 1) for Poiseuille, and have the form

v(x,t) =

( ) o v(~, t)

'

~(x, t) =

,(~,t)) ,,/(~, t)

~-(~,t)

///

"

X

+1 ~

///

,,,

_j -,

///

0

J

///

///

- 1

///

Poiseuille flow

Couette flow

Figure 3: Plane Couette and Poiseuille flows

The system for a Jeffreys-type fluid, which is to be solved in the interval I, reads as follows, Rev, - (1 - r

= r. - f,

(7"

at + ~ e - (I + a)Tv~: + #l(V~, T) = 0, ,~ "7, + ~ e + (1 --a)rv:~ +/32(v~, 7-) = O, r

~'+~-7-

(@ee

+(

l+a

2 ~-

1-a

2

))

" ~+#~(~'

(z2) r)

=

0,

with boundary conditions v ( - 1 , t ) = v(1,t) - 0, for Poiseuille flow, or v(0, t) = 0, v(1, t) = 1, for Couette flows.

(23)

The functions ~i, i - 1,2, 3 depend on the model. (See the description of different models in Section 1). The Oldroyd model corresponds to ~qi = 0, i = 1,2,3, from which we l+a - - ~ ' - ~ O". For the Giesekus model, one has deduce system (16) for (v, c~, r) where c~ = -7-7 #1 = ~ + ~ , #~ = ~ + ~ , ~ = (~ + z)~. Let (vs, a s , % , r s ) be a steady solution to system (22)-(23) (e.g., a Poiseuille flow or a Couette steady flow) and consider the linearized system around this state:

218

R e vt -

(1 -

e)v~

-

r. = 0,

a rOY, 4 Ov~ at + ~ee - ( 1 + a)(T,V~ + OX ) + ~-~0i~(-~X' %)Ui = 0, i=1

7

Ova. "-'0 4 Ov~ 7t + ~ee + (1 - a ) ( r s v , + r--~z ) + ~.~ i~/2(-~z , r,)ui = 0,

r Tt+Wee--

(@ee +(

1+ a

2

1-aa.))v~_ %-

(l+a

2

2 4

(24)

1-aa)Ov, 7-

2

Ox

OV s

= o, i--1

""

with homogeneous boundary conditions. Above 01, 02, 03 and 04 denote the partial derivatives with respect to the variables Ul =. v,, u2 = a, u3 = 7 and u4 = 7". Proceeding as in [47], one can show that the linear operator A associated with system (24) generates an analytic semi-group if 0 < e < 1 (non-zero Newtonian contribution), so that the implication "($3) ~ ($2)" is true in this case. Note that we only consider one-dimensional perturbations, but that the constitutive law is arbitrary. Following [47] we restrict now the study of stability to Oldroyd models (where/3~ = 0). It is easy to check that the steady Couette flow, solution of the steady equations corresponding to system (16)-(17), is given by ~k 2

vs(x)=x,

a , = W e ( l + k 2 ) ' r , = l + k 2' 0 < x < l ,

where k 2 = We2(1 - a 2 ) . Concerning the Liapunov (nonlinear) stability of the Couette flow under one dimensional perturbations, we have for instance the following result [47]. T h e o r e m 5.1 (Nonlinear stability to 1D p e r t u r b a t i o n s ) (i) Let e 6 (0, 8/9). Then the Couette flow is stable in H ~. (ii) Let e G (8/9, 1). Then there ezists some function k~(e) such that the Couette flow is stable in H 1 for all k < kl (e).

This result is proven by making use of an energy method. We refer to [47] for a proof and for other related results, e.g., sufficient conditions on e and k for unconditional stability in L 2. These results have been extended by Le Meur [50] to the case of multifluid flows. The situation for the plane Poiseuille flow for Oldroyd models is not as simple, as shown by the following result. T h e o r e m 5.2 (Existence of s t e a d y Poiseuille flows) (i) Let e 6 [0, 8/9). Then there exists a unique steady solution v~ = v~(x), which is very smooth on (0, 1), actually in the class of C ~ functions having continuous derivatives at all orders. 5 0 Let e 6 (8/9,1). that

Then there exists a critical Weissenberg parameter k~ > 0 such

(a) if k < k~, the conclusion of 5) holds; (b) if k > k~, there does not exist any C ~ solution, but a continuum of C o solutions, which are C ~ except at a finite number of points.

219

Some typical half profiles of Poiseuille flows are drawn on Figure 4.

o

0 < ~ < 8/9

v(x)

o

8/9 < e < 1

v(x)

E

k>k c

Figure 4: Profiles on (0,1) of the shear velocity for Poiseuille flows

Results of stability at small Reynolds numbers for the nonregular Poiseuille flows described in Theorem 5.2 are obtained in [48], where it is shown that the steady Poiseuille flows for which the velocity takes its values in the increasing part of the S-shaped curve (see Figure 5) only--excluding a neighbourhood of the max and the minmare nonlinearly stable. Stability holds when the perturbation from a steady state satisfies the following: the total shear stress is small in I-I1, the normal stresses are small in L a, and bounded pointwise by some large constant. Moreover, if Re is small enough, every unsteady Poiseuille flow converges, as t --~ oo, to some steady state, possibly nonregular, possibly unstable. The proofs of these results rely on the geometric study of the approximate dynamical system obtained at zero Reynolds number. This system has been thoroughly studied in [68], while [69] was devoted to a model system with only one equation for the stress. We go back to the linear stability of Couette flow of an Oldroyd fluid. The results are better understood if we draw (Figure 5) the curve of the total dimensional shear stress ~'(~/) versus the shear rate ~, = U/L, where U is the velocity of the upper plate and L the distance between the two parallel plates. We have the following stability result [47]. T h e o r e m 5.3 ( L i n e a r s t a b i l i t y of t h e C o u e t t e flow) (i) If 0 < e < 8/9, the Couette solution (v,, r~, a,) is linearly stable for all k 's. (ii) If 8/9 < ~ < 1, the Couette solution (vs, r~,e~s) is linearly stable if and only if 0 < k < k_ or k > k+, where k_ and k+ are the solutions of en 2 + (2 - 3~)n + 1 = O.

The stability regions are exactly those where the curve in Figure 5 is monotone, i.e.,

220

correspond to the values of/7 satisfying 0 < 4/ < ,~ or "} > "7~. The stability is lost when the spectrum of the linearized operator crosses the imaginary axis at O, eigenvalue of infinite multiplicity. The proof of Theorem 5.3 consists in showing that the underlying semi-group is analytic (because of the degeneracy of the equation for c~ and r in (22)), and then in localizing the spectrum by the Routh-Hurwitz criterion.

~*~

~=0

c < 8/9

=

9

> 8/9

,

!

E=I

2

Figure 5: Curves (-},'?*(-})) for different values of e E (0,1)

R e m a r k 5.1 It seems natural to conjecture that the linear stability regions in Theorem 5.3 coincide with the nonlinear stability regions. Theorem 5.1 gives the answer for part of the first increasing portion of the S-shaped curve in Figure 5. R e m a r k 5.2 The instability quoted in Theorem 5.3 is due to the aforementioned Sshaped curve. This constitutive instability--inherent to the model--has been questioned as unphysical by some authors. 5.4. N o n l i n e a r s t a b i l i t y Nonlinear stability results for viscoelastic fluids are very few. They essentially concern Jeffreys-type fluids. We have already mentioned those of [47] for the one-dimensional stability of Couette flows (see Section 5.3), and for the stability of flows of Jeffreys-type fluids which are small perturbations of the rest state (see Corollary 4.1). In a recent paper [70] Renardy has investigated the nonlinear stability of flows of Jeffreys-type fluids at low Weissenberg numbers. More precisely, assuming the existence of a steady flow (~, ~), he proves that this flow is linearly and Liapunov stable provided the spectrum of the linearized operator lies entirely in the open plane {~/~ < 0} and that the following quantity is sufficiently small We(1

+ (-1/2) sup (IVl + 0 B(V, f~

+ 01B( ,

221 where B(cr ~) = (~. V)~ +/3(V~, @), and 01 and 02 denote the partial derivatives of B with respect to re and ~ respectively.

5.5. Spectral studies As was mentioned before, no general theoretical result exists, which would justify the implication "($3) ~ ($2)', i . e . , the usual procedure of calculating the eigenvalues of the linearized system to assert the stability of viscoelastic flows. Despite this fact, a lot of studies have been devoted to such computations, and it is impossible to review them all. We have therefore selected significant examples, and quoted a few others. The general idea is to look for plane waves perturbations having the form e "t ek'x leading to spectral problems in a, which can be solved numerically. We emphasize once again that this kind of study does not imply the nonlinear--even the linear!--stability of the flow, if the assertion "($3) ~ ($2)", for instance, has not been proven mathematically. It is worth noticing that Tlapa and Bernstein [71] have proven that the Squire theorem holds true for the Poiseuille flow of an upper-convected Maxwell fluid. It means that any instability, which may be present for three dimensional disturbances, is also present for two dimensional ones at a lower value of the Reynolds number. This property is not true, in general, for non-Newtonian fluids [72]. Renardy and Renardy [66,73] have investigated the stability of plane Couette flows for Maxwell-type models involving the derivative (2). The flow lies between parallel plates at x = 0 and x = 1, which are moving in the y-direction with velocities -t-1, such as in Figure 6.

1

O

Y

Figure 6: Plane Couette flow with velocity +1 on the plates

The steady Couette solution has velocity v~ = (0, x, 0) and stress with components

or,=

(1 + a)W'e 1 l + k 2 , r s = l + k 2' % =

(1 -- a)We l+k 2 '

222

with k 2 = We2(1 - a2). One adds to the basic flow a perturbation for which the velocity has a component only in the z direction, v(x, y, z) = xey + r where here denotes a small parameter. The equations of motion and the constitutive relation are then linearized with respect to ~. To avoid some possible unphysical instabilities, the range of parameter a, which a priori belongs to ( - 1 , 1 ) , is restricted to the set {a >_ 1/2, k < 1 }. The first restriction ensures that the model is consistent with rod climbing, the later that one stays on the increasing part of the curve of the shear stress as a function of the shear rate, which has a maximum at k -- 1. The eigenvalue problem is solved numerically by the tau-Chebychev method [66]. It can be determined analytically that there is a continuous spectrum given by 1

=

1 - a2)We 2 [2(2 +

-

-
/3 > 0, for some constant/3.

q~Q,,wexh -]qllD[w]] -

(27)

227 Here, (-,-) denotes the scalar product in L 2 or 1[,~, and I" I the norm in these Hilbert spaces. For the approximation of r , we consider the finite element space: Th = {IT E T ; ITIK E Pk(K) 4, VK (5 Th}. In order to describe the approximation of the r equation of Problem (26) by the discontinuous Galerkin method of [100], we introduce the notation: 0 K - ( v ) = {x E OK, v ( x ) - n ( x ) < 0}, r e ( v ) ( x ) = lim r ( x + ev(x)) v...+O •

~

for x E OK-(v)

,

where OK is the boundary of K and n is the outward unit normal. We define the scalar products:

(T, IT)h -- E ('r, tT)K , KETh

(T'4"' IT4")hKv ~--T h f 0 '

K-(v) ~-~(v)- ~ ( v ) Iv 9n, d~,

where (., ")K denotes the scalar product in L2(K) or in L2(K). We also define the trilinear form bh on Xh x Th x Th by bh(v, r , a) = ((v- V ) r , IT)h + 1/2((div v) 1", a)h + ( r + - 7"-, IT+)h,vProblem (26) is approximated by the following system: Find (rh, vh,ph) E Th x Xh x Qh such that (rh, a)h + We (bh(vh, rh, IT) + (r

rh), a)h) - 2e (D[vh], IT)h = 0 VIT E Th,

(2s)

(rh, D[w])h + 2(1 -- e)(D[vh], D[w])h -- (Ph, divw)h = (f, w)h Vw E Xh, (divvh, q)h = 0 Vq E Qh6.3. A c o n v e r g e n c e t h e o r e m All known results are of the following type. If the continuous problem (26) admits a solution which is sufficiently smooth and small, then: 1. the approximate problem (28) (which is also nonlinear) admits a solution close to the continuous solution; 2. one gets an error estimate; 3. the approximate solution is unique and can be obtained by a fixed point iterative scheme. In other words a typical theorem is the following.

228 T h e o r e m 6.1 (Existence of c o n t i n u o u s and approximate flows. E r r o r estimate) Let e < 1 and k >_ 1. There exist two constants Co > 0 and ho > 0 depending on k, such that if Problem (26} admits a solution (v, r , p ) E t t k+2 • ][-Ik+l • H k+l satisfying M = m~x{ll v Ilk+2, II 1" IIk§ II P IIk+x} ~ Co, then, for all h < ho, Problem (~8} admits a solution (Vh, ~rh, Ph ) E Xh X Th • Qh, satisfying the following error estimate: there exists a constant C > 0 independent of h such that I~ - rhl + I D [ v - vh]l + Ip - phi < C h k+~/2.

Moreover, there exist two constants C~ > 0 and h'o > 0 such that M < C~ and h < h'o imply that the solution of Problem (28) is unique and is the limit of the fixed point iteration scheme (we omit the subscript h}" For aiven ( r n, v n , pn ), find( ,Tn+l, vn+l , pn+l ) E Th • Xh • Qh such that

( r ~+x, a) - 2, (D[vn+']), a) + We bh(v n, r ~+1 , a) = - W e (/3(Vv ~, v ~), a) V a e Th,

(29)

(r~+l,D[w]) + 2(1 - e)(D[v~+'], D[w]) - (p"+l,div w) = (f,w) Vw e Xh, (div v ~+~, q) = 0 Vq e Qh. The proof of the first part of this theorem is given in [101] for k = 1, and in [102] for k > 2. Uniqueness and fixed point aspects are studied in [103].

6.4. Miscellaneous r e m a r k s 1. An analogous result is vahd for continuous approximations of 7" when upwinding is performed by the streamline upwinding Petrov-Galerkin method (SUPG) [104]. The same is true for finite element methods based on a quadrangular mesh [105]. 2. For cost reasons, k is most often equal to one in practical implementations. Commonly used finite elements are: (a) the Hood-Taylor finite element: /92 for the velocity, continuous P1 for the pressure, and discontinuous P1 for the stress; (b) P2 plus bubble (also called P+) for the velocity, discontinuous Px for the pressure, and discontinuous P2 for the stress; (c) on a quadrangular mesh, (image of) Q2 for the velocity, discontinuous Q1 for the pressure, and discontinuous Q2 (or even incomplete Q2 with eight nodes) for the stress. 3. When We = 0, the Oldroyd-B model (26) reduces to a three-field version of the Stokes problem. For e < 1, this problem is stable under condition (27). It was proven in [106] that, in the case of the Maxwell-type problem (where e = 1), one has to add a second inf sup condition to obtain stability:

229

inf ~ex.

sup (t r, D [ w ] ) > 7 > 0 for some constant 7~ T . I~l ID[w]l -

(30)

As explained in [1061 condition (30) may be satisfied either by imposing D[Xh] C Th, suggesting the use of discontinuous Th, or by giving a sufficient number of interior nodes in each K of Th in case of continuous Th. This last fact was first observed numerically in [98], where a sixteen node Q1 finite element approximation of r is used. For efficiency reasons it seems preferable to use finite element satisfying (30) even for e < 1, but near one. Inexpensive finite elements satisfying (27) and (30) have been recently developed (see [107-109]). Most of the above comments apply to three dimensional problems, but corresponding 3D simulations are still exceptional for cost reasons. 4. Theorem 6.1 is still valid for models with several relaxation times, as well as for models with a quadratic 13 in r , such as the Giesekus and the linearized PhanThien-Tanner models. (See [110]) 5. For White--Metzner-type models, the associated Stokes problem obtained for We = 0 is then nonlinear. This is related to quasi-Newtonian models and is studied in [111]. Numerical analysis for We > 0 has not been done yet. 6. Nothing seems to be known concerning the numerical analysis of Maxwell-type mod-

els. 7. Problem (26) can be viewed as a transport equation in r for given v, and a Stokes system in (v, p) for given 1-. The fixed point iteration scheme described in Theorem 6.1 does not use this fact. A more natural iterative scheme, which uncouples the r and the v equations, reads as follows: For given (7"~, v ~, p'~), find (r~+l , vn+l , pn+l) E Th x X h X Q h such that ( r ~+1 , or) + We bh(v ~, T n+l , tT) = - W e (fl(Vv ", r~), (r) + 2e (D[v~], ~r) Vet E Th, 2(1 - e + 7)(D[v~+~], D[w]) - (p~+X, divw) = ( f , w ) + 27(D[v"], D [ w ] ) - (rn+l,D[w]) Vw e Xh,

(31)

(divv TM, q) = 0 Vq E Qh, where g > 0 is a fixed parameter. The convergence of this iterative scheme is proven in [112]. 8. Most of the numerical analyses so far concern steady flows. For unsteady problems, the convergence of the following scheme has been proven in [113]"

230

For given (r", v'~, p=), find (v"+l,v"+i,p n+l) E Th x Xh x Qh such that Re ((v ~+1 - v")fiSt, w) + 2(1 - e)(D[vn+l], D[w]) - (pn+~, div w) +(vn+',D[w]) = (f(t,+i),w) Vw E Xh,

We((7 "n+l

-

-

r")l,St, ,,-) +

we

bh(v", I" TM,a) + (r ''+l, a')

(32)

-2e (D[v"+'], tr) = - W e (fl(Vv", ~'"), tr) Vtr E Th, (divv '~+1, q) = 0 Vq e Qh, where (St denotes the time step. This result is extended in [114] to an uncoupled unsteady scheme based on (31). 7. Conclusion Mathematical and numerical analyses of differential models for viscoelastic fluid flows are highly challenging domains, which still need a lot of effort. However, significant progress has been made during the last decade, and mathematical results have shown to be quite useful for the modelling. Classification of differential models with respect to typical properties such as change of type or loss of evolution has been done. Robust results--it means that they are independent of the models--for existence of regular flows have been obtained in different situations: slow steady flows, or steady flows perturbing a uniform flow; unsteady flows on a short time interval, or unsteady flow for all times, but small data; flows with inflow boundary values for certain simple geometries, ... The important problem of stabihty of viscoelastic flows is far more involved than for Newtonian flows, and many problems still remain open. However, it is clearly established that the difficulty lies in the relationship between various mathematical notions of stability. Some results have been obtained in this direction for restricted classes of flows and/or of models. Moreover several important studies of spectral stability have been performed. Last, numerical analyses of schemes for approximating simple flows have been developed: the convergence results give some confidence in the most often used methods for the computation of flows in realistic situations. A c k n o w l e g m e n t s . We thank H. Le Meur for his valuable comments during the preparation of this paper and for his drawings of all the figures. REFERENCES 1. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1988. 2. R.B. Bird and J.M. Wiest, Constitutive equations for polymeric liquids, Ann. Rev. Fluid Mech., 27 (1995) 169-193. 3. T.H. Kwon and S.F. Shen, A unified constitutive theory for polymeric liquids, I and II, Rheol. Acta, 23 (1984) 217-230, and 24 (1985) 175-188. 4. C. Guillop6 and J.-C. Saut, Mathematical analysis of differential models with internal variables for viscoelastic fluids, in preparation.

231 B.J. Edwards and A.N. Beris, Remarks concerning compressible viscoelastic fluid models, J. Non-Newtonian Fluid Mech., 36 (1990) 411-417. I.M. Rutkevich, Some general properties of the equations of viscoelastic incompressible fluid mechanics, J. Appl. Math. Mech. (PMM), 33 (1969) 30-39. I.M. Rutkevich, The propagation of small perturbations in a viscoelastic fluid, J. Appl. Math. Mech. (PMM), 34 (1970) 35-50. D.D. Joseph, M. Renardy and J.-C. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluids, Arch. Rat. Mech. Anal., 87 (1985) 213-251. D.D. Joseph and J.-C. Saut, Change of type and loss of evolution in the flow of viscoelatic fluids, J. Non-Newtonian Fluid Mech., 20 (1986) 117-141. 10. F. Dupret and J.-M. Marchal, Loss of evolution in the flow of viscoelastic fluids, J. Non-Newtonian Fluid Mech., 20 (1986) 143-171. 11. Y. Kwon and A.I. Leonov, Stability constraints in the formulation of viscoelastic constitutive equations, J. Non-Newtonian Fluid Mech., 58 (1995) 25-46. 12. C. Guillop~ and J.-C. Saut, Mathematical problems arising in differential models for viscoelastic fluids, in Mathematical Topics in Fluid Mechanics, J.F. Rodrigues and A. Sequeira (eds.), Longman Scientific and Technical, Pitman, 1992, 64-92. 13. D.D. Joseph, J. Matta and K.P. Chen, Delayed die swell, J. Non-Newtonian Fluid Mech., 24 (1987) 31-65. 14. A.B. Metzner, E.A. Uebler and Chang Man Fong, Converging flows of viscoelastic materials, AIChE J., 15 (1969) 750-758. 15. L.E. Fraenkel, On a linear partly hyperbolic model of viscoelastic flow past a plate, Proc. Roy. Soc. Edinburgh, 114 A (1990) 299-354. 16. M.J. Crochet and V. Delvaux, Numerical simulation of inertial viscoelastic flows with change of type, in Nonlinear Evolution Equations That Change Type, B.L. Keyfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, SpringerVerlag, Berlin, 1991, 47-66. 17. M. Renardy, A well-posed boundary value problem for supercritical flow of viscoelastic fluids of Maxwell-type, in Nonlinear Evolution Equations That Change Type, B.L. Keyfitz and M. Shearer (eds.), IMA Volumes in Mathematics and its Applications 27, Springer-Verlag, Berlin, 1991, 181-191. 18. D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, Springer-Verlag, Berlin, 1990. 19. B.D. Coleman and W. Noll, Foundations of linear viscoelasticity, Rev. Modern Phys., 33 (1961) 239-249. 20. J.-C. Saut and D.D. Joseph, Fading memory, Arch. Rat. Mech. Anal., 81 (1983) 53-95. 21. J.S. Ultman and M.M. Denn, Anomalous heat transfer and a wave phenomenon in dilute polymer solutions, Trans. Soc. Rheol., 14 (1970) 307-317. 22. G. Schleiniger, M.C. Calderer and L.P. Cook, Embedded hyperbolic regions in a nonlinear model for viscoelastic flow, in Current Progress in Hyperbolic Systems: Riemann Problems and Computations, W.B. Lingquist (ed.), Contemporary Mathematics 100, American Mathematical Society, Providence, 1990. 23. C. Guillop6 and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., Th. Meth. Appl., 15 (1990) 849-869. .

.

232 24. M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech., 65 (1985), 449-451. 25. A. Hakim, Analyse math~matique de modules de fluides visco~lastiques de type White--Metzner, Th~se de l'Universit~ Paris-Sud, Orsay, 1989. 26. C. Guillop~ and J.-C. Saut, Existence and stability of steady flows of weakly viscoelastic fluids, Proc. Roy. Soc. Edinburgh, 119 A (1991) 137-158. 27. R. Talhouk, Analyse math~matique de quelques ~coulements de fluides visco~lastiques, Th~se de l'Universit~ Paris-Sud, Orsay, 1994. 28. R. Talhouk, Ecoulements stationnaires de fluides visco~lastiques faiblement compressibles, Comptes Rend. Acad. Sc. I, 320 (1995) 1025-1030. 29. M. Renardy, Inflow boundary conditions for steady flows of viscoelastic fluids with differential constitutive laws, Rocky Mount. J. Math., 18 (1988) 445-453, and 19

(1989) 561. 30. M. Renardy, An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions, J. Non-Newtonian Fluid Mech., 36 (1990) 419-425. 31. M. Renardy, Existence of steady flows of viscoelastic fluids of Jeffreys-type with traction boundary conditions, Diff. Int. Eq., 2 (1989) 431-437. 32. M. Renardy, Existence of steady flows for Maxwell fluids with traction boundary conditions on open boundaries, Z. Angew. Math. Mech. 75 (1995) 153-155. 33. M. Tajchman, Conditions aux limites absorbantes pour des fluides visco~lastiques de type diff~rentiel, Th~se de l'Universit~ Paris-Sud, Orsay, 1994. 34. B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 139 (1977) 629-651. 35. L. Halpern, Artificial boundary conditions for incompletely parabolic perturbations of hyperbolic systems, SIAM J. Math. Anal., 87 (1985) 213-251. 36. E.J. Hinch, The flow of Oldroyd fluid around a sharp corner, J. Non-Newtonian Fluid Mech., 50 (1993) 161-171. 37. M. Renardy, The stress of an upper-convected Maxwell fluid in Newtonian field near a re-entrant corner, J. Non-Newtonian Fluid Mech., 50 (1993) 127-134. 38. M. Renardy, A matched solution for corner flow of the upper-convected Maxwell fluid, J. Non-Newtonian Fluid Mech., 58 (1995) 83-89. 39. A. Hakim, Mathematical analysis of viscoelastic fluids of White-Metzner type, J. Math. Anal. Appl, 185 (1994) 675-705. 40. R.E. Gaidos and R. Darby, Numerical simulation and change of type in the developing flow of a nonlinear viscoelastic fluid, J. Non-Newtonian Fluid Mech., 29 (1988) 59-79. 41. M. Renardy, Local existence of the Dirichlet initial boundary value problem for incompressible hypoelastic materials, SIAM J. Math. Anal., 21 (1990) 1369-1385. 42. M. Renardy, Initial value problems with inflow boundaries for Maxwell fluids, SIAM J. Math. Anal., (1996) to appear. 43. R. Talhouk, Unsteady flows of viscoelastic fluids with inflow and outflow boundary conditions, Appl. Math. Letters, (1996) to appear. 44. J.U. Kim, Global smooth solutions for the equations of motion of a nonlinear fluid with fading memory, Arch. Rat. Mech. Anal., 79 (1982) 97-130.

233 45. J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., 3 (1959) 120-122. 46. A. Valli, Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method, Ann. Sc. Norm. Sup. Pisa, 10 (1983) 607-647. 47. C. Guillop6 and J.-C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, Math. Mod. Anal. Numer., 24 (1990) 369-401. 48. D.S. Malkus, J.A. Nohel and B.J. Plohr, Analysis of a new phenomenon in shear flow of non-Newtonian fluids, SIAM J. Appl. Math., 51 (1991) 899-929. 49. M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical Problems in Viscoelasticity, Longman Scientific and Technical, Burnt Mill, Harlow, 1987. 50. H. Le Meur, Existence, unicit6 et stabilit6 de plusieurs types d'6coulements de fluides visco61astiques avec interface, Th~se de l'Universit6 Paris-Sud, Orsay, 1994. 51. H. Le Meur, Existence locale de solutions des 6quations d'un fluide visco61astique avec fronti~re libre, Comptes Rend. Acad. Sc. I, 320 (1995) 125-130. 52. J.P. Tordella, Unstable flow of molten polymers, in Rheology: Theory and Applications 5, F. Eirich (ed.), Academic Press, New York, 1969. 53. N. E1 Kissi and J.-M. Piau, The different capillary flow regimes of entangled polydimethylsiloxane polymers: macroscopic slip at the wall, hysteresis and cork flow, J. Non-Newtonian Fluid Mech., 37 (1990) 55-94. 54. J.-M. Piau, N. E1 Kissi and B. Tremblay, Influence of upstream instabilities and wall slip on melt fracture and sharkskin phenomena during silicones extension through orifice dies, J. Non-Newtonian Fluid Mech., 34 (1990) 145-180. 55. R.G. Larson, Instabilities in viscoelastic flows, Rheol. Acta, 31 (1992) 213-263. 56. M. Renardy and Y. Renardy, Pattern selection in the B6nard problem for a viscoelastic fluid, Z. Angew. Math. Mech., 43 (1992) 154-180. 57. M. Avgousti and A.N. Beris, Viscoelastic Taylor-Couette flow: bifurcation analysis in the presence of symmetries, Proc. Roy. Soc. London A, 443 (1993) 17-37. 58. G. Iooss, Bifurcation et stabilit6, Publications Math6matiques d'Orsay 31, 1974. 59. A.F. Neves, H. de Souza Ribeiro and O. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal., 67 (1986) 320-344. 60. M. Renardy, On the type of certain Co semi-groups, Comm. Part. Diff. Eq., 18 (1993) 1299-1307. 61. M. Renardy, On the linear stability of hyperbolic partial differential equations and viscoelastic flows, Z. Angew. Math. Phys., 45 (1994) 854-865. 62. M. Renardy, A rigorous stability proof for plane Couette flow of an uppe-convected Maxwell fluid at zero Reynolds number, Eur. J. Mech. B, 11 (1992) 511-516. 63. V.A. Gorodtsov and A.I. Leonov, On a linear instability of a plane parallel Couette flow of a viscoelastic fluid, J. Appl. Math. Mech. (PMM), 31 (1967) 310-319. 64. M. Renardy, On the stability of parallel shear flow of an Oldroyd B fluid, Diff. Int. Eq., 6 (1993) 481-489. 65. V.A. Romanov, Stability of plane-parallel Couette flow, Funct. Anal. Appl:'7 (1993), 137-146.

234 66. M. Renardy and Y. Renardy, Stability of shear flows of viscoelastic fluids under perturbations perpendicular to the plane of flow, J. Non-Newtonian Fluid Mech., 32 (1989) 145-155. 67. G. Prodi, Teoremi di tipo locale per il sistema di Navier-Stokes e stabilitk delle soluzioni stazionare, Rer.d. Sem. Mat. Univ. Padova, 32 (1962) 374-397. 68. J.A. Nohel, R.L. Pego and A.E. Tzavaras, Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid, Proc. Roy. Soc. Edinburgh, 115 A (1990) 39-59. 69. J.A. Nohel and R.L. Pego, Nonlinear stability and asymptotic behaviour of shearing motions of a non-Newtonian fluid, SIAM J. Math. Anal., 24 (1993) 911-942. 70. M. Renardy, Nonlinear stability of flows of Jeffreys fluids at low Weissenberg numbers, Arch. Rat. Mech. Anal. 132 (1995) 37-48. 71. G. Tlapa and B. Bernstein, Stability of a relaxation type viscoelastic fluid with slight elasticity, Phys. Fluids, 13 (1970) 565-568. 72. F. J. Lockett, On Squire's theorem for viscoelastic fluids, Int. J. Eng. Sci., 7 (1969) 337-349. 73. M. Renardy and Y. Renardy, Linear stability of plane Couette flow of an upperconvected Maxwell fluid, J. Non-Newtonian Fluid Mech., 22 (1986) 23-33. 74. B.J.A. Zielinska and Y. Demay, Couette-Taylor instability in viscoelastic fluids, Phys. Rev. A, 38 (1988) 897-903. 75. S.J. Muller, R.G. Larson and E.S.G Shaqfeh, A purely elastic transition in TaylorCouette flow, Rheol. Acta, 28 (1989) 499-503. 76. R.G. Larson, E.S.G. Shaqfeh and S.J. Muller, A purely viscoelastic instability in Taylor-Couette flow, J. Fluid Mech., 218 (1990) 573-600. 77. E.S.G. Shaqfeh, S.J. Muller and R.G. Larson, The effects of gap width and dilute solution properties on the viscoelastic Taylor-Couette instability, J. Fluid Mech., 235

(1992) 285-317. 78. R.G. Larson, S.J. Muller and E.S.G. Shaqfeh, The effect of fluid rheology on the elastic Taylor-Couette instability, J. Non-Newtonian Fluid Mech., 51 (1994) 195-225. 79. P. Laure, H. Le Meur, Y. Demay and J.-C. Saut, Linear stability of multilayer plane Poiseuille flows of Oldroyd-B fluids, Pr6publication 96-12, Universit~ Paris-Sud, Math~matiques, submitted. 80. K.P. Chen, Elastic instability of the interface in Couette flow of viscoelastic liquids, J. Non-Newtonian Fluid Mech., 40 (1991) 261-267. 81. C.H. Li, Stability of two superposed elastoviscous liquids in plane Couette flow, Phys. Fluids, 12 (1969) 531-538. 82. Y. Renardy, Stability of the interface in two-layer Couette flow of upper-convected Maxwell liquids, J. Non-Newtonian Fluid Mech., 28 (1988) 99-115. 83. Y.Y. Su and B. Khomami, Interfacial stability of multilayer viscoelastic fluids in slit and converging channel die geometries, J. Rheol.. 36 (1992) 357-387. 84. S.A. Orszag, Accurate solution of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50 (1971) 689-703. 85. Y.Y. Su and B. Khomami, Numerical solution of eigenvalue problems using spectral

235 techniques, J. Comp. Phys., 100 (1992) 297-305. 86. S. Yiantsios and B.G. Higgins, Analysis of superposed fluids by the finite element method: linear stability and flow developement, Int. J. Numer. Meth. Fluids, 7 (1987) 247-261. 87. T.A. Herbert, Die neutrMe Fls der ebenen Poiseuille-StrSmung, Habilitation, Universits Stuttgart, 1977. 88. Y. Saad, Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices, Lin. Alg. Appl., 34 (1980) 269-295. 89. S. Scotto, Calcul num~rique des modes critiques de l'6quation d'Orr-Sommerfeld visco~lastique, Rapport de DEA, Universit~ de Nice Sophia-Antipolis, 1994. 90. R. Sureshkumar and A.N. Beris, Linear stability of viscoelastic Poiseuille flow using an Arnoldi based orthogonalisation algorithm, J. Non-Newtonian Fluid Mech., 56 (1995) 151-182. 91. K.P. Chen, Interracial instability due to elastic stratification in concentric coextrusion of two viscoelastic fluids, J. Non-Newtonian Fluid Mech., 40 (1991) 155-175. 92. K.P. Chen and D.D. Joseph, Elastic short wave instability in extrusion flows of viscoelastic liquids, J. Non-Newtonian Fluid Mech., 42 (1992) 189-211. 93. K.P. Chen and Y. Zhang, Stability of the interface in coextrusion flow of two viscoelastic fluids through a pipe, J. Fluid Mech., 247 (1993) 489-502. 94. A. 0ztekin and R.A. Brown, Instability of a viscoelastic fluid between rotating parallel disks: analysis for the Oldroyd-B fluid, J. Fluid Mech., 255 (1993) 473-502. 95. N. Phan-Thien, Cone and plate flow of the Oldroyd-B fluid is unstable, J. NonNewtonian Fluid Mech., 13 (1985) 325-340. 96. D.O. Olagunju, Asymptotic analysis of the finite cone-and-plate flow of a nonNewtonian fluid, J. Non-Newtonian Fluid Mech., 50 (1993) 289-303. 97. K.P. Chen, The onset of elastically driven wavy motion in the flow of two viscoelastic liquids films down an inclined plane, J. Non-Newtonian Fluid Mech., 45 (1992) 21-45. 98. J.-M. Marchal and M.j. Crochet, A new finite element for calculating viscoelastic flows, J. Non-Newtonian Fluid Mech., 26 (1987) 77-114. 99. M. Fortin and A. Fortin, A new approach for the FEM simulation of viscoelastic flows, J. Non-Newtonian Fluid Mech., 32 (1989) 295-310. 100P. Lesaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor (ed.), Academic Press, New-York, 1974, 89-123. 101J. Baranger and D. Sandri, Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds I - Discontinuous constraints, Numer. Math., 63 (1992) 13-27. 1022. Baranger and D. Sandri, High order finite element methods for the approximation of viscoelastic fluid flow, Proceedings of the Tenth International Conference on Computing Methods in Applied Sciences and Engineering, R. Glowinski (ed.), Paris, 1992, 185-194. 103J. Baranger and D. Sandri, Some remarks on the discontinuous Galerkin method for the finite element approximation of the Oldroyd-B model~ submitted.

236 104J). Sandri, Finite element approximation of viscoelastic fluid flow: existence of approximate solutions and error bounds. Continuous approximation of the stress, SIAM J. Numer. Anal., 31 (1994) 362-377. 105A. Bahar, J. Baranger and D. Sandri, Quadrilateral finite element approximation of viscoelastic fluid flow, Rapport de l'6quipe d'analyse num6rique Lyon-Saint-Etienne 162 (1993), submitted. 106.M. Fortin and R. Pierre, On the convergence of the mixed method of Crochet and Marchal for viscoelastic flows, Comput. Meth. Appl. Mech. Engrg., 73 (1989) 341-350. 107.V. Ruas, An optimal three field finite element approximation of the Stokes system with continuous extra-stresses, Japan J. Ind. Appl. Math., 11 (1994) 103-130. 108.V. Ruas, J.H. Carneiro and M.A. Silvaramos, Approximation of the three-field Stokes system via optimized quadrilateral finite elements, Math. Mod. Anal. Num., 27 (1993) 107-127. 109.D. Sandri, Analyse d'une formulation k trois champs du probl~me de Stokes, Math. Mod. Anal. Num., 27 (1993) 817-841. l l0J. Baranger and D. Sandri, Finite element method for the approximation of viscoelastic fluid flow with a differential constitutive law, First European Computational Fluid Dynamics Conference, Bruxelles, 1992, C. Hirsch (ed.), Elsevier, Amsterdam, 1993, 1021-1025. 111J. Baranger, K. Najib and D. Sandri, Numerical analysis of the three-field model for a quasi-Newtonian flow, Comput. Meth. Appl. Mech. Engrg., 109 (1993) 281-292. 112.K. Najib and D. Sandri, On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow, Num. Math., 72 (1995) 223-238. 113J. Baranger and S. Wardi, Numerical analysis of a finite dement method for a transient viscoelastic flow, Comput. Meth. Appl. Mech. Engrg., 125 (1995) 171-185. 114.S. Wardi, Convergence of an uncoupled algorithm for a transient viscoelastic flow, submitted.


237

C o m p u t a t i o n of 2D viscoelastic f l o w s for a differential constitutive e q u a t i o n Y. Demay Institut Non Lin6aire de Nice, 1361 route des Lucioles, 06560 Valbonne, France. 1. I N T R O D U C T I O N We will consider in the following chapter numerical difficulties encountered in the numerical simulation of two-dimensional viscoelastic flows described by a differential constitutive equation. As a particular emphasis is put on numerical simulation of molten polymer flows existing in rheological laboratory experiments or polymer processing, only the finite elements method is considered. Our goal is to help the unfamiliar reader to understand the considerable mathematical and numerical difficulties encountered and we refer to several reviews for more detailed studies. Let us first recall some mathematical considerations concerning fitting of numerical methods to general properties of the considered equations.

1.1. The type of an equation and links with modelling It was recalled that the mathematical type of a system of equations is characterized by notion of characteristics associated with a direction. If all the characteristics are complex (resp. real) the problem is elliptic (resp. hyperbolic). EUipticity and hyperbolicity are the mathematical notions associated with diffusion and propagation. The reader is possibly not familiar with these mathematical notions, but he is surely used to the consequences. Ellipticity has a regularizing effect on a singularity while hyperbolicity propagates it. If a phenomenon is controlled by an elliptic system of equations, effects at long distance depend only on averaged quantifies. Let us give some examples. For an elastic rod loaded at an extremity, stress and displacement values far enough from this extremity are only determined by mean values of loading (Saint-Venant principle). In the same way the flow of a Newtonian fluid at the exit of a pipe (Poiseuille flow) depends only on the flow rate at the entry and the velocity profile at exit is regular no matter what it is at entry. Other examples include those with thermal diffusivity. On the contrary, for a phenomenon controlled by a hyperbolic system of equations, shocks and singularities are transported. Several examples such as shockwaves, water waves or interfaces of immiscible (and non reacting) fluids are familiar. Let us come back to the example of pipe

238 flow. For a Newtonian fluid, the Poiseuille flow is fully developped (in stress and velocity) at a distance of two diameters of the entry. As shown previously, viscoelastic effects are introduced through convected derivatives of the viscoelastic part of the extra stress tensor and as a consequence, the entrance length is largely increased. This is in good agreement with experiments as can be seen on the experimental and numerical flow birefringence pattern of Fig. 40 in III.2. The problem now, for a viscoelastic fluid, is that this transport equation is also able to transport shocks and, by the effect of equilibrium equations, shocks in extra stress severely pollute the velocity field. This is why the introduction of a geometrical singularity, such as the re-entrant comer, induces stress singularity and hence both mathematical and numerical problems. In the previous chapter, the mathematical specification of viscoelastic flows governed by a differential law was given out for a general class of models including the Oldroyd-B or WhiteMetzner model. Viscoelasticity introduces a coupling between a transport system of equations and a Navier-Stokes equation. The problem was then to study the mathematical nature of this set of equations. It was found in section 2.2, that for the steady creeping flow of a Maxwell fluid, both types of characteristics exist. Furthermore the characteristics associated with the vorticity equations can change of type for some values of the Mach number M =Re We. In the following we will restrict our study to low Reynolds number.

1.2. Realistic geometries The mathematical existence of the viscoelastic flow for an Oldroyd B fluid was proved in chapter II for steady flows and for sufficiently small data by theorem 3.1. It was proved for unsteady flows and locally in time by theorem 4.1. In both cases the frontier of the domain is assumed to be smooth enough (smoothness required by definition of Hs/2(Of2) space in theorem 3.1 and C 3 in theorem 4.1). In industrial processes as well as in laboratory experiments, molten polymer flows in complex geometries. The polymer is generally molten in a screw extruder, then distributed in a flat die and stretched in air (cast film process), or templated in air (tube), or pushed in a mould (injection moulding) and finally cooled and solidified. Various kinematical or geometrical singularities, such as re-entrant corners or change of boundary conditions at extrusion, are present. These processes are commonly used in industry and a mathematical simulation are relatively easily performed with a Newtonian fluid at low Reynolds number (due to the high viscosity, the Reynolds number in polymer processing is low). We will point out in the following that viscoelasticity is to be introduced carefully in such a geometry. More precisely, classical viscoelastic constitutive equations, such as the upper convected Maxwell or Jeffreys models, used to analyse rheological experiments on a cone-plate apparatus introduce considerable numerical problems.

239

1.3. The Finite Element Method The finite element method was fast introduced (in the 60s) to compute numerically elastic deformations of solids. It can be easily proved that this system is elliptic. As a consequence of no particular space direction, boundary conditions on displacement or stress vector are given on the whole frontier of the domain. In the 70s, this method was extended to computation of purely viscous flow by a convenient treatment of the compressibility condition. In this case again, the system is elliptic in the velocity components and hence boundary conditions on the velocity field or stress vector are given on the boundary. These conditions can be, for example, vanishing velocity components on some part of the boundary (if the fluid sticks at the wall) or vanishing stress vector on another part (for a free surface with air). Mixed conditions involving a component of velocity and the associated condition on a component of the stress vector can also be used on a symmetry axis. The important point is that on each point of the boundary two and only two convenient conditions are associated with two-dimensional Stokes flow. Unless for large value of Reynolds number, this is not changed by introduction of inertia which adds derivative of lower order. Because of the success encountered by finite elements in the solution of elliptic problems, it was extended (in the 80s) to the advection or transport equation which is a hyperbolic equation with only one real characteristic. This equation can be solved naturally for an analytical velocity field by solving a time differential equation. It appeared important, when the velocity field was numerically obtained, to be able to solve simultaneously propagation and diffusion equations at low cost. By introducing upwinding in test functions or in the discretization scheme, the particular nature of the transport equation was considered. In this case, a particular direction is given at each point (the direction of the convecting flow) and boundary conditions are only considered on the part of the boundary where the flow is entrant. 1.4. The computation of a viscoelastic flow: a mathematical problem So far, numerical techniques for a purely hyperbolic (transport) or a purely elliptic equation are clearly established and the problem is now to manage with a mixed type problem. As shown in section 3.4.2 of the previous chapter, the natural problem of boundary conditions mathematically associated with this system of equations is difficult to solve rigorously. In fact, in the considered geometry, it is assumed that established extra stress tensor values are given in the entry section of the boundary and velocity or stress vector are given on the whole frontier. Due to nonlinearity, it was not realistic to consider a rigorous finite element method introducing an adapted treatment of each characteristic. For example, results of Renardy ((16), (32) of previous chapter) are obtained for small perturbations of a uniform flow. As the numerical

240 method was not able to take into account the particular nature of the system of equations governing a viscoelastic flow, it was split into an elliptic problem for velocity and pressure more or less coupled with an advection equation in the extra stress. It is important to notice that this splitting technique is present in all numerical methods, at least through a particular treatment of the constitutive equation and the choice of test functions. Very soon it appeared that significant nurnerical difficulties are observed in realistic geometries such as convergent or extrusion geometries and generally convergence with mesh refinement was not obtained. A non closed controversial debate began to decide if the failure was due to weak or non adequate numerical approximation, to the mathematical nature of this system of equations unable to accept geometrical singularities or to the use of a non-realistic viscoelastic law. Simultaneously, more sophisticated approaches were developed in all of these ways. The approximation space was improved and the nature of the viscoelastic part of the constitutive equation was modified at high values of the stress. Up to now the situation is not completely clear, despite the great number of studies devoted to this subject by authors from a large range of fields such as applied mathematics, mechanical engineering or physics. In recent years, important progress has been made in the numerical simulation of viscoelastic fluid flows for increasing elasticity. Very detailed reviews have been devoted to numerical computation of viscoelastic flows ([ 10], [24] and [39]). In the following sections, we recall the main mathematical results concerning the numerical solution of the Stokes problem (16)-(14) by a finite element (Galerkin) method. 2. C O M P U T A T I O N OF A PURELY VISCOUS F L O W The main difficulty is to conveniently satisfy the incompressibility condition. In the following we will first recall the continuous mathematical formulation of the Stokes problem. Then it is recalled that a compatibility condition between pressure and velocity elements is necessary to prove convergence. Finally several possible strategies to solve the discretized system are developed. 2.1. The w e a k f o r m u l a t i o n of the Stokes problem

Let us consider the following Stokes problem on domain s with boundary conditions on O~ = FlkgF2 (with FlnF2=O) (1)

o = 2nD(U)- p Id,

(2)

div(c) + F = 0 ,

(3)

div(U)=0,

(4)

U=Uo

x ~ U1.

c.n=O x

E

F2.

241 where n is a vector normal to the boundary at x. Multiplying equations (2) and (3) respectively by test functions V (vanishing on F1) and q, and integrating by parts, we obtain the following "weak" problem: (5)

ITID(U):D(V)- I V p . V = IF.V

(6)

.[ q div(U) - 0 f~

Equations (5)-(6) hold for any test function V in a subspace of Hl(f~) including homogeneous boundary conditions on 1"1 and for any q in the L2(f~) space. A continuous inf-sup inequality ensures the existence and uniqueness of the solution of the weak problem (5)-(6). The considered sohtion is a saddle point (minimum in V and maximum in q) of the Lagrangian: (7)

1

L(V,q) - ~ .f TID(V):D(V)- j'F.V - J" q div(V) . f2 ~ f~

2.2. The numerical solutions 2.2.1. The elements If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the discrete problem has a unique solution converging to the solution of the continuous one (see equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked dement by element. Finite dement methods for viscous flows are now wen established and pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two strategies are used to compute the numerical solution of these equations involving velocity and pressure. 2.2.2. The penalty method (8)

Let us define the augmented Lagrangian as: =1 r )2 Lr(V,q) ~ ~ rlv D(V):D(V)- ~F.V- ~ qdiv(V) + ~ I div(V f~ f~ f~ f~

The solution is also a saddle point of the augmented Lagrangian: ~Lr ~gLr (9) ~ ( U , p ) = 0 , ~(U,p)=0 In order to solve equations (9), a generalization of the Uzawa algorithm is generally chosen for its simplicity: 1

(10) ~ I rlv D(un)'D(V) + r f div(Un) div(V)= ~F.V + f pn-1 div(V) f~ f2 f~ f~ (11) pn = pn- 1 + r div(U n)

242

Equation (10) holds for any function V vanishing on F1. The last term of the augmented Lagragian (for r----0,Lr is a Lagrangian) introduces a penalty of the incompressibility condition and the Uzawa algorithm allows us to satisfy equation (3) as precisely as we wish using moderate values of r. 2.2.3. The two-field solution The linear system obtained by the discretization of equations (5)-(6) can also be solved directly. Notice that this system is symmetric but not definite positive. A three-field version of the Stokes problem was considered in [ 17] and a second inf-sup condition is then necessary to obtain stability (equation (30) of 6.4). 2.3. Effect of a n o n - s m o o t h geometry

We first recall some basic properties of Newtonian flows in two classical singular geometries. 2.3.1. The stick-slip singularity

.

.

.

.

.

.

.

.

.

.

0

Figure 1.a: The angular sector f2 of angle co and boundary F1 and F2 Let us consider as flow domain, the angular sector f~ of angle co and boundary F1 and F2 as represented in Fig. 1.a). The boundary of the domain is regular except at point O. The boundary conditions are vanishing velocity on F1 (the fluid sticks to the wall on F1), vanishing normal velocity and tangential stress vector on F2 (the fluid slips on the wall on F2). The particular case of the half plane (co = x) is referenced as the stick-slip problem and represents a local analysis of a fluid slipping on the wall downstream of point O. This problem has been largely studied theoretically for a Newtonian fluid. It was proved that solutions of poor regularity (presenting infinite values of pressure and viscous stress) at discontinuity exist. These singular solutions have the form (in polar coordinates): A

(12) U(r,0)= r a U(O), p(r,0)= r a-1 ~(0), and ct is computed as a solution of an eigenvalue problem for an elliptic equation in 0 (0_1. 1.

For to = ~, ct = ~ is a solution. These solutions can be considered as similarity solutions, eigenvector of the dilation r --~ ar. 2.3.2. The re-entrant comer singularity Let us now consider the same flow domain (represented in Fig. 1.a) with the boundary conditions of vanishing velocity on F] and 1-'2 (the fluid is sticking at the wall on F1 and F2). This problem too has been largely studied for a Newtonian fluid. In this case, singular solutions of the homogeneous Stokes problem exist if t~ is a solution of the following equation: (14) (sin(otto))2 = (sin(to))2 Otto

CO

It can be easily verified that there is at least one solution of equation (14) satisfying 0

S

S = ~yr = ~

-ozz = N2 ,

(28) where S is the surface of the specimens. Obviously, both conditions cannot be satisfied simultaneously. Figure 10 shows a comparison of FN/S, -N1 = N2-N3 and N2, N2 and N3 being determined from the birefringence m e a s u r e d in the yz and x-z planes according to the linear stress optical rule. It a p p e a r s clearly t h a t the best a g r e e m e n t is obtained with the second normal s t r e s s difference, which m e a n s t h a t the free surface conditions hold r a t h e r on the x-y t h a n on the y-z surface. It should be noticed t h a t the values of birefringence m e a s u r e d in the different planes were homogeneous within 10 % in the whole specimen, except close to the free surfaces normal to the x direction, w h e r e end effects were present. The difference observed at small strains in Fig. 10 m a y be a t t r i b u t e d to a small compressive force arising w h e n the specimen is placed b e t w e e n t h e plates of the rheometer, and which has not entirely relaxed at the beginning of the experiment. -(2),4 003

-0,3

UJ n," I-- -02.

/

J

,._.1

~

m~m

0'01

_

~ -.

F i g u r e 10. FN/S, N2 and -N1 as a function of s h e a r s t r a i n during a creep test (see text). (~) FN/S; (w) N2; and ( . ) - N 1 .

. @ . . _ . 1 4, I

~ ,..~

@@

~ |

|

3

|

4

SHEAR STRAIN

d. Decomposition of stress: The comparison between the time evolution of stress and birefringence during an elongational test at constant s t r a i n r a t e suggests t h a t the total tensile stress is m a d e up of two contributions: (i) an entropic contribution arising from chain orientation and (ii) a second contribution r e l a t e d to the

271 initial stress value, which is strongly temperature dependent and vanishes at t e m p e r a t u r e s h i g h e r t h a n Tg + 20 ~ If we assume t h a t the entropic contribution is related to the birefringence through the equilibrium stressoptical law, we can calculate its time evolution from that of the birefringence and the curves in Fig. 5. This has been done for stretching experiments carried out at 98.5 ~ and 0.05 s -1 for PS and 142 ~ and 0.025 s -1 for PC. ~20 ft. v

O9 15 (/)

0,0

Figure 11. Total (n,,u) and entropic p a r t ( o , o ) of the tensile stress as a function of ~2_ ~-1 for PS (re,o) and PC (u,o) (see text).

0,5

1,0

1,5

2,0 2,5 ~2~-1

3,0

In Fig. 11 the total stress and its entropic part calculated from a have been plotted for PS and PC as a function of the quantity ~2 _ ~-1, where ~ is the extension ratio L/L0. An almost linear behaviour is found for both PS and PC for the entropic part of the stress, up to values of ~ - 2. A neo-Hookean shear modulus can be calculated as the slope of these curves: values of approximately 1 and 3 MPa are found for PS and PC, respectively. Although these values are higher th a n the respective plateau moduli of the two polymers (i.e. 0.2 MPa for PS and 1 MPa for PC), their ratio is close to t h a t of the plateau moduli. Actually, the neo-Hookean moduli defined from the entropic part of the stress are closer to the values at the crossover between the G'(co) and G"(~) curves corresponding to the high frequency limit of the rubbery plateau (i.e. 0.5 MPa for PS and 2 MPa for PC). The non-entropic contribution to the stress appears in Fig. 11 as the difference between total and entropic stress. It is found t h a t the n o n - e n t r o p i c s t r e s s r e m a i n s a l m o s t c o n s t a n t d u r i n g the s t r e t c h i n g experiment, apart from the initial overshoot observed only at temperatures very close to the Tg. An a t t e m p t has been made to describe the particular stress-optical behaviour observed close to Tg. The idea was to associate the entropic part of the stress with relaxation times corresponding roughly to the rubbery plateau and the terminal zone (see Fig. 12), whereas the non-entropic part is assumed to be related to shorter time relaxation phenomena (glass transition and glassy state). This approach is similar to t hat proposed by Inoue et al. [34] who considered two contributions to the stress with different associated stressoptical coefficients.

272

10

10

A

4

%

.

.

.

.

0 log(co)

Figure 12. Master curves for G' (m) and G"(~) of PS at 173 ~ Full lines r e p r e s e n t the c o n t r i b u t i o n of long relaxation times associated with entropic stress.

n

0=5

1,0

1,5 ~-a= (MPa)

2,0

Figure 13. Full lines: model predictions for the data of Fig. 9 (see text).

An upper-convected Maxwell model h a s been used w i t h t h e full relaxation spectrum for the calculation of the stress, but for calculating the birefringence this spectrum has been restricted to long relaxation times as shown in Fig. 12. The model predictions for the data of the Fig. 9 are shown in Fig. 13. The deviations from the linear stress-optical rule are well accounted for by this very simple model. However, the model does not describe the stress d a t a in simple elongation, and in p a r t i c u l a r the initial s t r e s s v a l u e s at t e m p e r a t u r e s close to the Tg.

3. 2. Complex flows a. Experimental techniques: The abrupt contraction is a very interesting flow situation, which allows one to characterize simultaneously shear flow near the d o w n s t r e a m die wall, elongational flow along the centreline, mixed flow in the converging section and exit effects. Fig. 14 shows a typical experimental set-up for a p l a n a r contraction [18, 23]. The die with transparent glass walls is continuously fed by an e x t r u d e r . It consists in a large r e s e r v o i r in which p r e s s u r e a n d t e m p e r a t u r e may be measured, and a final adjustable slit section, which m a y be changed to modify the flow geometry. One important problem in such a design is to ensure t h a t the flow in the die can be considered as two-dimensional. Wales [6] showed t h a t the perturbation introduced by the presence of the die walls was negligible as soon as the aspect ratio of the die (width over depth) was greater t h a n 10. Moreover, this influence is more pronounced for isoclinic fringes t h a n for isochromatics.

273

F i g u r e 14. E x p e r i m e n t a l set-up for slit die birefringence (from [18]). Table 2 shows t h a t the m a j o r p a r t of e x p e r i m e n t a l devices u s e d in the l i t e r a t u r e r e s p e c t s t h i s o r d e r of m a g n i t u d e , w h i c h allows a good c o m p r o m i s e b e t w e e n q u a l i t y of results a n d facility of observation. Table 2 D i m e n s i o n s (in ram) of c o n t r a c t i o n flow e x p e r i m e n t s from t h e l i t e r a t u r e Reservoir (width x depth) 10 x 12

Die l a n d (width x depth) 10 x 2

Aspect ratio (width/depth) 5

Contraction ratio 6

Ref.

30 x 10

10 x i 20 x 1 30 x 1

10 20 30

10 10 10

[6] [6] [6]

19.9 x 25.4

19.9 x 1.99

10

12.8

[13]

7.6 x 15

7.6 x 1

7.6

15

[15]

9.5 x 14.8

9.5 x 2.5

3.8

5.9

[23]

25.4 x 10.2

25.4 x 2.54

10

4

[25]

16 x 20

16 x 2.5

6.4

8

[18]

24 x 8

24 x 2

12

4

[26]

45 x 20

45 x 2

22.5

10

[17]

57.2 x 5.08

57.2 x 1.27

45

4

[8]

10 x 3

10 x 1.2

8.3

2.5

[27]

14.7 x 14.7

14.7 x 2.3

6.4

6.4

[28]

[24]

274 The results presented in sub-sections 3.2.b and 4 were obtained for a 8/1 contraction, using the experimental set-up described in Fig. 14 (die land: 16 x 2.5 ram; aspect ratio: 6.4), with a sodium monochromatic light (wavelength: x = 0.589 ~m) and different die lengths (5 and 20 ram) and converging angles (45 and 90 o). b. Experimental results: These results will be focused on the flow of a linear polyethylene (LLDPE Lotrene FC 1010) previously characterized (Chapter II-I). In Fig. 15 are presented the general changes observed in patterns of isochromatic fringes when increasing the flow rate. When looking at a particular picture, different areas may be distinguished: the reservoir: upstream of the contraction, the stress level is minimal. When moving downstream along the centreline, we observe the first order fringe, then the second, third and so on, with quasi-circular shapes. - the restriction: the birefringence level is very high here. On the centreline, birefringence reaches a maximum just before the land entry and t h e n decreases. In the perpendicular direction, the order of the fringes continues to increase, to reach a m a x i m u m in the corner. This value is however very difficult to determine due to the very high number of fringes. - the die channel: the order of the fringes decreases when the polymer relaxes along the channel. It may be seen on some pictures t h a t the zeroth order fringe is not observed before the exit, which means t h a t the stress relaxation is not achieved during these flows. the exit: a drastic change is observed just a few millimetres before the die exit, with an abrupt local increase, followed by a total relaxation in the extrudate. A global quantification in terms of stresses of such a p a t t e r n would require the knowledge of the extinction angle x everywhere, to apply the expressions given in sub-section 2.2.b (Eqs. 22 and 23). In fact, t he d e t e r m i n a t i o n of x using the set-up presented in Fig.14 is very difficult, because, due to side effects, isoclinic fringes are very wide and imprecise [5]. For th a t reason, it is easier to follow the change in birefringence along the centreline, where its value is directly proportional to the normal stress difference (Eq. 21). An example corresponding to the different pictures of Fig. 15 is given in Fig. 16. W h en increasing the flow rate, the n u m b e r of fringes increases regularly. The stress build-up in the reservoir appears sooner and t he relaxation in the channel is lower at high flow rate. The general evolution along the centreline remains similar, whatever the flow rate, and the m a x i m u m value measured at the entry of the downstream flow increases very regularly (Figures 16 and 17). -

-

c. Validity of the stress optical law: The stress optical law has been carefully checked by Wales [6] in steady s h e a r conditions for a wide n u m b e r of molten polymers. However, the experimental set-up limited the range of stresses to a maximum of 105 Pa. In elongational situations, a departure from the linearity was previously observed

275

Figure 15. Change in birefringence patterns with the flow rate (LLDPE Lotrene FC 1010, 205 ~

276

60 tO

o v

50

X

LU o

40

LJ.J 0

3O

n"

20

z

Z

Li.. ILl rr"

10

10

0

10

20

AXIAL POSITION (mm)

Figure 16. Change in birefringence along the centreline (Lotrene FC 1010, 205 ~ u 20 s -1, 9 84 s -1, o 133 sq).

A

o

T~

60

50

Os

j O

10

X

u.J o z I.LJ Z

LL. ILl CC

~9 J

40 30 / ~

20

/

10

/ !

I

0

50

" ....

I

'~'

100

150

APPARENT SHEAR RATE (s-l)

Figure 17. Change in maximum birefringence on the centreline with the apparent shear rate (Lotrene FC 1010, 205 ~ for polystyrene, above 1 MPa by Matsumoto and Bogue [29] and 2 MPa by Muller and Froelich [19] (see also Fig. 5 and sub-section 3.1.b). At h i g h e r stresses, the n u m b e r of fringes in an a b r u p t contraction makes it very difficult to determine the birefringence level, except along the centreline. It could thus be interesting, as proposed by Beaufils et al. [18], to e v a l u a t e the m a x i m u m value, m e a s u r e d j u s t before the die entry, and to compare it to the corresponding theoretical value of oI- (~II, computed using a

277 flow simulation software. This method allows one to define correctly the order of magnitude of the stress optical coefficient, but its main drawback is that the result will depend on the constitutive equation chosen for the simulation. Effectively, as explained in Chapter III.2, it involves also the validity of the chosen constitutive equation and the computation method. However, the results obtained are similar to those issued from more specific techniques. For example, for the LLDPE studied here (Lotrene FC 1010), we obtained values r a n k i n g from 2.07 10 -9 m2/N (K-BKZ) to 2.21 (Phan-Tien Tanner), 2.57 (Generalized Oldroyd-B) and 2.88 (Inelastic Carreau-Yasuda). In Fig. 18 are presented the results obtained for generalized Oldroyd-B and multimode PTT simulations (see Chapter III-2). It can be seen that, in this transient elongational situation at high strain rate (the elongational strain rate can reach approximately 120 s-l), the proportionality between birefringence and stress difference is only valid up to 0.2 MPa. 60 0.-,0

~ o X

v

W Z

LU (.9 Z n"

LL LU EE

50

~

/O ~00

40

f

9 ,0

30

/

20

/

10

f

O0

o~

/ '

....

I

1

'

.

I

' '.

2

I

3

PRINCIPAL STRESS DIFFERENCE

'

"'

4

(X 10 5 Pa)

Figure 18. Validity of the stress optical law for the LLDPE at 205 ~ (o" PTT, e" Generalized Oldroyd-B). 4. APPLICATION TO COMPLEX FLOW STUDIES 4.1 I n f l u e n c e o f d i e g e o m e t r y

In Fig. 19 are presented, for the same flow rate, the isochromatic patterns obtained for different channel geometries (long (20 ram) and short (5 mm) dies, 45 ~ and 90 ~ converging angles). If the shape of the fringes appears different for the 45 ~ convergent, the change along the centreline is very similar: stress build-up is identical, and the relaxation mechanism also, until reaching the proximity of the die exit, where it becomes very rapid for the shorter dies. It is clear that this similarity is only valid along the centreline. Other flow parameters (velocity field, streamlines ..) are obviously dependent on die entry geometry.

278

Figure 19. Change in birefringence patterns with the die geometry (Lotrene FC 1010, 145 ~

~/a = 36 s-l).

279

Figure 20. Change in birefringence patterns with the flow rate (LDPE FN 1010,175 ~

280

4.2 Influence of molecular s t r u c t u r e We study here the case of a branched polyethylene (LDPE FN 1010), characterized by long chain branching, whose rheological behaviour has been previously described in Chapter II-1. Compared to linear LLDPE, it exhibits s t r a i n - h a r d e n i n g in elongational situations and higher values of first normal stress difference. Figure 20 presents birefringence patterns obtained for a long die (20 ram) at 175 ~ If the birefringence along the centreline changes similarly to the preceding cases, the general aspect of the pattern is different, principally around the r e - e n t r a n t corner. Characteristic W shapes are observed for the LDPE, whereas LLDPE exhibited more regular evolution. These shapes result in a non-monotonic variation of birefringence between the centreline and the die walls (see Chapter III-2). The area affected by this particular behaviour is c o n c e n t r a t e d a r o u n d the c h a n n e l e n t r y and develops p r o g r e s s i v e l y downstream when increasing the flow rate. For the long die, the pattern far downstream appears regularized even at high flow rate, whereas, for the short die, the perturbed area progresses until the die exit. Similar observations were made by Checker et al. [15] on a polypropylene and explained by a high level of elasticity combined with a strain hardening elongational behaviour, which is also the case for the LDPE. 4~3 Flow ~n.c~abflities We have seen that flow birefringence is a powerful tool for studying steady and transient flows of molten polymers. This technique can also be used to assess local flow conditions during flow instabilities. These problems are specifically treated in Chapter III-4. We would just like to present here the potential of birefringence techniques for such studies. Tordella [35] was probably the first to show HDPE a n d LDPE birefringence patterns highly perturbed after the onset of flow instabilities. Discontinuities in the extinction bands and non stationary patterns were then observed, which was confirmed l a t e r by Vinogradov et al. [24] on polybutadienes or Oyanagi et al. [36] on HDPE and PS. Piau et al. [17] observed on polybutadiene slight pulsations at the die exit, correlated to the frequency of the sharkskin defect.

Figure 21. Discontinuities in the birefringence pattern in the die land at high flow rate (LDPE FN 1010, 145 ~

89 = 170 s-l).

281 We can see in Fig. 21 that, for the LDPE (FN 1010) above a given flow rate, bands of discontinuity appear in the birefringence pattern. These bands are observed parallel to the flow direction, at an intermediate position between centreline and die wall. Such "defects" may also be detected in the converging flow near the die entry (Fig. 22). For these flow conditions, the extrudate begins to exhibit distorded volume shapes, characteristic of melt fracture. By comparison with rheological studies carried out on a capillary rheometer, we may assume that these defects belong to the family referred to as "upstream instablities" in Chapter III.4.

Figure 22. Discontinuity bands at the die entry (LDPE FN 1010, 145 ~

~/a = 287 s "1)

Fig. 23 illustrates another type of unstable behaviour for a polystyrene. In this case, an alternative discharge of the large entry vortices in the main flow leads to a periodic asymmetric pattern, both in reservoir and die land, resulting in an helical type defect at the die exit. Such observations were also reported by Oyanagi et al. [36]. 5. CONCLUSION We presented some applications of the birefringence techniques for the characterization of orientation and stress field in both simple and complex deformations. The validity of the stress optical law in these particular flow situations was checked and assessed. Birefringence techniques appear to be very powerful for the study of polymer melt processing and for the understanding of polymer flow behaviour.

282

F i g u r e 23. Oscillating sequence for a polystyrene (PS Gedex 1541, 180 ~

89 = 200 s -1)

283

REFEPd~qCES Io

2. 3. 4. 5. .

.

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28 29. 30. 31.

D. Brewster, Trans. Roy. Soc. London (1816) 156 M.E. Mach, "Optisch-Akustische Versuche", Calve, Prague (1873) A.S. Lodge, Nature, 176 (1955) 838 W. Philippoff, Nature, 178 (1956) 811 M.E. Mackay and D.V. Boger, in: "Rheological Measurement", A.A. Collyer, D.W. Clegg Eds., Elsevier, London (1988) J.L.S. Wales, "The Application of Flow Birefringence to Rheological Studies of Polymer Melts", Delft University Press (1976) P.L. Frattini and G.G. Fuller, J. Rheol., 28 (1984) 61 S.R. Galante and P.L. Frattini, J. Non Newt. Fluid Mech., 47 (1993) 289 I.M. Ward (ed.), Structure and Properties of Oriented Polymers, Applied Science Publishers, London (1975) W. Kuhn and F. Grtin, Koll. Zeit., 101 (1942) 248. L.R.G. Treloar, Trans. Faraday Soc., 50 (1954) 881. O. Kratky, Koll. Zeit., 64 (1933) 213. C.D. Han, "Rheology in Polymer Processing", Academic Press, New York (1976) H. Janeschitz-Kriegl, "Polymer Melt Rheology and Flow Birefringence", Springer Verlag, Berlin (1983) N. Checker, M.R. Mackley and D.W. Mead, Phil. Trans. R. Soc. London, A 308 (1983) 451 S.T.E. Aldhouse, M.R. Mackley and I.P.T. Moore, J. Non Newt. Fluid Mech., 21 (1986) 359 J.M. Piau, N. E1 Kissi and A. Mezghani, J. Non Newt. Fluid Mech., 59 (1995) 11 P. Beaufils, B. Vergnes and J.F. Agassant, Int. Polym. Proc., 2 (1989) 78 R. Muller and D. Froelich, Polymer, 26 (1985) 1477. J.A. Van Aken, F.H. Gortemaker, H. Janeschitz-Kriegl and H.M. Laun, Rheol. Acta, 19 (1980) 159. K. Osaki, S. Kimura and M. Kurata, J. Polym. Sci., Polym. Phys. Edn 19, (1981) 151. R.M. Kannan and J.A. Kornfield, Rheol. Acta, 31 (1992) 535. G. Sornberger, J.C. Quantin, R. Fajolle, B. Vergnes and J.F. Agassant, J. Non Newt. Fluid Mech., 23 (1987) 123 G.V. Vinogradov, N.I. Insarova, B.B. Boiko and E.K. Borisenkova, Polym. Eng. Sci., 12 (1972) 323 S.A. White and D.G. Baird, J. Non Newt. Fluid Mech., 29 (1988) 245 H.J. Park, D.G. Kiriakidis, E. Mitsoulis, K.J. Lee, J. Rheol., 36 (1992) 1563 M. Van Gurp, C.J. Breukink, R.J.W.M. Sniekers and P.P. Tas, SPIE, 2052 (1993) 297 R. Ahmed and M.R. Mackley, J. Non Newt. Fluid Mech., 56 (1995) 127 T. Matsumoto and D.C. Bogue, J. Polym. Sci. Polym. Phys. Ed., 15 (1977) 1663 W. Retting, Colloid Polym. Sci., 257 (1979) 689. R. Wimberger-Friedl and J.G. De Bruin, Rheol. Acta, 30 (1991) 419.

284 32. 33. 34. 35. 36. 37.

L.R.G. Treloar, The Physics of Rubber Elasticity, Clarendon Press, Oxford, 1949. R. Muller and J.J Pesce, Polymer, 35 (1994) 734. T. Inoue, H. Okamoto and K. Osaki, Macromolecules, 24 (1991) 5670. J.P. TordeUa, J. Appl. Polym. Sci., 7 (1963) 215 Y. Oyanagi, K. Kubota and K. Ohji, 4th PPS Meeting, Orlando (1988) M.R. Mackley and I.P.T. Moore, J. Non Newt. Fluid Mech., 21 (1986) 337


285

Comparison between experimental data and numerical models J. Guillet, C. Carrot, B.S. Kima J.F. Agassant, B. Vergnes, C. Bdraudob J.R. Clermont, M. Normandin, u B~reaux c a Laboratoire de Rh~ologie des Mati~res Plastiques, Facultd de Sciences et Techniques, Universitd Jean Monnet, Saint-Etienne 23 rue du Docteur Paul Michelon, 42023 Saint-Etienne Cedex 2, France b CEMEF, Ecole des Mines de Paris, URA CNRS 1374 BP 27, 06904 SOPHIA-ANTIPOLIS Cedex, France c Laboratoire de Rhdologie, URA CNRS 1510 Domaine Universitaire, BP n ~ 53 X, 38041 GRENOBLE Cedex 9, France

1. INTRODUCTION A complete understanding of flow phenomena occurring in entry and exit regions of complex geometries is still elusive at the present time for viscoelastic fluids. Understanding entry and exit flows of non-Newtonian fluids like polymer melts is of importance in polymer processing operations, such as extrusion and injection moulding. Typically, Newtonian fluids flow radially through the entry region in a contraction geometry, sweeping out the corners of the upstream channel and exhibiting small size vortices. They give a constant extrudate swell ratio for the jet emerging from the downstream channel. At the opposite, large corner vortices may appear for many viscoelastic fluids. Concerning the extrudate jet emerging from the die, the swell ratio is not only higher than that corresponding to a Newtonian fluid, but also depends on flow conditions, such as the range of strain rates, the temperature and the length-to-diameter ratio of the die. An illustrative example of the previous considerations may be given for polyethylene melts. It is admitted that low density polyethylene (LDPE) melts develop rapid vortex growth in an abrupt contraction, and that high density polyethylene (HDPE) and linear low density polyethylene (LLDPE) melts do not. However, in exit flows, all these polyethylene melts can swell notably, and, for many years, there has been no clear understanding about differences in entry and exit flows of these polymer melts.

286 Observation, discussion and u n d e r ~ d i n g of recircttlating vortices in flows of polymer solutions and melts through abrupt contraction geometries have been the subject of many papers over the last 30 years. (e.g. review papers by Boger [1], White et al. [2], McKinley et al. [3]) The question of the presence, origin and growth of vortex remained unanswered until the works by Cogswell [4] and White and Kondo [5]. In both papers, it was suggested that materials involving an increasing extensional viscosity versus the extensional rate exhibit vortex enhancement, whereas vortices are small or absent for materials of constant or decreasing extensional viscosity. In further papers [6-7], the magnitude of the transient elongational stress growth was emphasized to be the most significant, whether vortices occur or not. The combined effects of strain rate and molecular characteristics such as molecular weight distribution and long chain branching [8-9] may imply quite different behaviour for the stresses through the entrance of the contraction. Thus, it would not be unusual to observe differences in entry flow patterns of polymers of a given type. This definitive conclusion may also be considered as relevant to the exit flow behaviour of molten polymers, since it is generally admitted t h a t the extrudate swell phenomenon originates in the main from extensional strain recovery. From a numerical viewpoint, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the numerical problem, in relation to the nonlinearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry. In the story of numerical flow simulation, the ability to predict observed and significant viscoelastic flow phenomena of polymer melts and solutions in an abrupt contraction has been unsuccessful for many years, in relation to the incomplete rheological characterization of materials, especially in elongation. The numerical treatments have often been confined to flow of elastic fluids with constant viscosity, described by differential constitutive equations as the Upper Convected Maxwell and Oldroyd-B models. Fortunately, the recent possibility to use real elastic fluids with constant viscosity, the so-called Boger fluids [10], has narrowed the gap between experimental observation and numerical prediction [11]. Various numerical studies indicated a loss of convergence at relatively low values of the Weissenberg number. The so-called ~high Weissenberg number problem" relates the inability to obtain numerical solution for flows in complex geometries involving singularities, for high or even moderate values of the Weissenberg number [ 12]. In particular, it was found t h a t the greater the mesh was refined, the lower the maximum Weissenberg number for convergence. Several research groups have concentrated on the problem of computational instability. The failure of calculations has often been attributed to approximation errors, which refer to the inability of the numerical method to fit the set of governing equations. So far, mlmerical studies have revealed that the usual computational instability, occurring with simple but unrealistic models, was overcome to a great extent by the use of new integral models, which have enabled

287 more realistic Weissenberg numbers and extrudate swell ratios to be reached [1316]. Nevertheless, it was shown recently that the convergence problems were at least partially due to the use of a constitutive equation which did not exhibit a steady solution at the Weissenberg number attempted, rather than to numerical difficulties [17]. In pioneering work, Keunings and Crochet [18] used the Phan-Thien Tanner model, with an additional purely viscous retardation term to correct the unrealistic rapid viscosity decrease predicted by this model in simple shear flow. The improvement of convergence with a refined mesh was found to be opposite to that observed in previous numerical studies. This improved convergence might be due to the Phan-Thien Tanner model itself or to the additional retardation term, which appears to stabilize the solution. It should be pointed out that the improvement of convergence might also be related to realistic preditions of shear and elongational viscosities by the PhanThien Tanner model, when compared to the Upper Convected Maxwell, Oldroyd-B and White-Metzner models. Satisfactory numerical results were also obtained with multi-mode integral constitutive equations using a spectr~_~m of relaxation times [7, 17, 20-27], such as the K-BKZ model in the form introduced by Papanastasiou et al. [ 19]. Although different numerical approaches were used by these research groups, the calculations revealed large vortex growth of LDPE with increasing Weissenberg number, similar to that observed experimentally in an axisymmetric abrupt contraction. However, the differences in vortex growth of LDPE and HDPE related to their elongational behaviour, presented in the paper by Luo and Mitsoulis [7] have to be pointed out. Their computed results indicate an increasing vortex size and intensity versus the Weissenberg number for strain-thickening extensional viscosity of the melt under consideration. On the other hand, for a fluid which exhibits a strain-thinning extensional viscosity, the weak vortex size and intensity observed at low flow rates are insensitive to the increase of the Weissenberg number. In addition, the authors came to the conclusion that strainhardening apparently enhances the vortex, but is not responsible for its presence or absence in the way that the level of elasticity is. Feigl and Ottinger [28] presented an extensive comparison of numerical and experimental results for the entry flow problem. They used a Rivlin-Sawyer integral constitutive equation, with a spectrum of relaxation times and a non-zero constant ratio of the second to first normal stress difference. The general performance of the model was tested by comparing the numerical predictions of vortex growth and entrance pressure loss at different flow rates with the experimental results related to the axisymmetric contraction flow of a LDPE melt reported by Knobel [29]. The numerical results were found to be consistent with experimental results and in some cases, in quantitative agreement. A good consistency was obtained for the vortex size and the increase of shear and extensional strain rates in the upstream channel. However, small or even large discrepancies still remained concerning the increase of the entrance pressure loss with flow rate and the evolution of shear strains in the flow geometry. Birefringence measurements are often performed to compare theoretical and experimental stress distributions in an abrupt contraction (see Section III-1). Such comparisons have been already published for the White-Metzner [30], KBKZ [27, 31] and Wagner [32, 33] constitutive equations. Generally speoking,

288 stress increase in the contraction entry is well described, but the relaxation along the downstream channel is often more difficult to capture. Considering now the extrudate swell problem, a few papers deal with the flow simulation of memory integral fluids at the exit of long and short dies [22-25, 27, 34-35]. Some of the papers previously cited present comparisons between experimental and numerical results. Goublomme et al. [23] presented numerical results for the extrudate swell problem for a long capillary die. In their calculation, they used a K-BKZ constitutive equation involving a spectrum of relaxation times and a damping function, either of the Wagner [36] or of the Papanastasiou type [19]. To avoid the rapid viscosity decrease of the model at high shear rates, the authors introduced a weak Newtonian contribution, ass,~med to be equivalent to the effects of the smallest relaxation times of the true spectr~m, which are not available because they relate very few data accessible at very large frequency values in dynamic experiments. If the calculation provided consistent numerical results for a long capillary die, the assumption of a short die, which required the inclusion of the upstream converging section in the calculations, resulted in largely overestimated values of the swell ratio. The numerical data presented in a second paper [24] proved that, if extrudate swell was practically unaffected by nonisothermal flow conditions, it was considerably modified by change of parameters in the K-BKZ equation. A non-zero constant ratio of the second to first normal stress difference led to significantly lower computed extrudate swell ratios towards the experimental data, as a consequence of modification of the extensional behaviour of the fluid. The experimental and numerical results reported by Luo and Tanner [22] are of particular interest since they are related to a thoroughly characterized IUPAC LDPE melt. In order to ensure good predictions in mixed flow situations, they relaxed the separability assumption in the K-BKZ form adopted in their study, at least in elongation, allowing a parameter of the damping function to vary according to each relaxation time of the memory function. Predictions close to experimental data were obtained at low shear rates (~ = 1 s-l), even with very short dies. However, when increasing the shear rate, the m~merical results were found to be larger than experimental ones. As pointed out later by Goublomme et al. [23], neither the non-isothermal flow assumption, nor the arbitrary decrease of normal stress ratio can affect the numerical swell predictions, to eliminate completely the discrepancy between numerical and corresponding experimental swell ratios. It should also be underlined that the calculated extrudate swell was found to be very sensitive to the wall slip boundary condition. This states a problem for the choice of a suitable criterion for wall slippage. A similar study is reported in the recent paper by Barakos and Mitsoulis [25]. It is worth mentioning that simulations for the same IUPAC LDPE melt have been undertaken, using the same constitutive equation with p a r a m e t e r s determined by Luo and Tanner [22]. Comparison are made with extrusion experiments given by Meissner [37]. Concerning calculations for short and long dies, the trends are similar to those observed by Luo and Tanner [22]. If the agreement is generally good for short dies and low shear rates, meaningful overestimation becomes larger and larger as the die length-to-diameter ratio and the apparent shear rate increase. It should also be noted that quantitative discrepancies still remain between calculated and experimental entrance pressure drops (or Bagley end corrections).

289 In the purpose of comparisons between theory and experiments, this short review points out the need for significant experimental studies related to flow p a t t e r n s , pressure drops and extrudate swell ratios for polymer melts characterized in shear and elongation. Streamlines, velocity components and stress fields related to flow birefringence studies should be used to validate numerical simulations of entry and exit flows. Progress in flow computations in domains involving the upstream and downstream channels together with the exit region are also necessary. In order to get a better insight into these major issues, the purpose of the present work is to report some experimental and numerical results on flows in planar and axisymmetric contractions for two polyethylene melts, namely a linear low-density polyethylene (LLDPE) and a low-density polyethylene (LDPE). Various constitutive equations are set up to model more or less thoroughly the rheological properties in simple flow, such as a memory-integral Wagner-type model, and the differential multimode Phan-Thien Tanner (m FI~) and generalized Oldroyd-B (GOB) models. From a numerical viewpoint, comparisons with experiments will be made between results provided by two numerical methods, the first one involving a discontinuous Galerkin finite element method [38, 39] and the second one considering the stream-tube analysis developed by Clermont [40]. 2. MATERIALS The polymers used in the present work are two commerdal polyethylenes, supplied by Elf Atochem and previously characterized (see Section II - 1) : a linear low density polyethylene (LLDPE FC 1010) and a low density polyethylene (LDPE FN 1010). The LLDPE material is an ethylene-butene copolymer, containing 4 % of butene. The LDPE polymer is known to involve long chain branching while LLDPE has a r a t h e r low content of branches, which are known to be short. These structures explain why large differences could be expected in their rheological behaviour in shear and elongation and, consequently, in complex flow situations. The molecular characteristics of both samples are given in Section II - 1. 3. CONSTITUTIVE EQUATIONS The description of flow patterns in entry and exit flows requires one to provide an adequate mathematical and mechanical description of the behaviour of polymer melts, by means of realistic constitutive equations. These equations must describe the melt behaviour from linear viscoelasticity conditions to nonlinear flow situations. Though it should be emphasized that the constitutive equation chosen for any process should provide the information required, particular rheological equations of state are generally considered as more realistic for polymer solutions and melts. These models involve integral forms of constitutive equations of the KBKZ type, the advantage of which lies in their connection to molecular aspects of viscoelasticity. With this type of equation, numerical calculation are often difficult to perform, due to the particle tracking problem and computing time requirements. These difficulties can be overcome to a great extent using stream-tube analysis.

290 Differential equations of the Oldroyd-type are easier to i m p l e m e n t for flow computation. According to these considerations, we selected an integral constitutive equation of the Wagner-type, and two differential constitutive equations, namely the multimode Phan-Thien Tanner (mPTT) and a generalized Oldroyd-B (GOB) models. It should be noted that two of these models, the Wagnertype and the mPTT, predict bounded extensional viscosities as well as shearthinning viscosities, whatever the strain rate is; conversely, the GOB model predicts infinite extensional viscosity in the long time range. The LLDPE and LDPE polymers have been thoroughly characterized in simple shear and extensional flows in a previous part of this book (see Section II 1). To enhance the u n d e ~ d i n g of the present section, we recall rapidly the basic features of these models.

3.1. T h e W a g n e r - t y p e i n t e g r a l constitutive equation Derived from the Lodge's rubberlike liquid theory, the Wagner model is based on a concept of separability, since it is assumed that the memory function is the product of a time-dependent linear function by a strain-dependent nonlinear function. W h e n considering an increasing deformation, this model, which is a particular case of the B K Z model, is written as follows : t

m(t- 1:). h(Ii,I2). (~ t (1:)d2:

g(t) =

(1)

-oo

where m ( t - z) is the memory function of the linear viscoelasticity, h(II,I 2) the -1 damping function and (~ t is the Finger relative strain tensor. 3.1.1. Memory function From the concept of separability, the memory function of the linear viscoelasticity is required. This memory function can be related to a discrete relaxation time spectra, m, available from dynamic experiments, given by: N re(t-z) = Z ~ exp(" ( t - z ) ) (2) i-1 ~ ~'i where 1~i and ~i are the viscosities and the discrete relaxation times, respectively. These parameters were calculated for the two polyethylenes, at 160 ~ using a procedure given by Carrot et al. [41], which enables the recovery of a rnirlimnm number of relaxation times (Table 1). Using these values, the loss modulus G" and the elastic modulus G' versus frequency curves can be recovered with maximum errors lower t h a n 4 % (see Section II- 1).

291

Table 1 Discrete relaxation time spectra of LDPE and LLDPE at T = 160 ~ ,

, ,

_

,

~*i[S]

,

LLDPE ,

, ~i

.

.

.

.

.

-

,

[Pa.s]

LDPE

, ,

. . . . ki[s]

_ _

,

.

~i [pa.s]

1.28 E-04

2.367 E+02

6.45 E-04

9.810 E+01

6.12 E-03

1.346 E+03

5.35 E-03

2.855 E+02

4.10 E-02

3.363 E+03

2.85 E-02

7.679 E+02

2.77 E-01

4.691 E+03

1.55 E-01

2.619 E+03

2.01 E+00

3.726 E+03

8.91 E-01

6.464 E+03

1.57 E+01

2.007 E+03

4.58 E+00

1.219 E+04

1.35 E+02

9.563 E+02

2.34 E+01

1.325 E+04

1.18 E+02

7.296 E+03

3.1.2. D_~mping function Different forms have been proposed for the damping function but m a n y of them are not acceptable for both shear and elongation. In this work, we retained for the damping function a modified form already proposed by Soskey et al. [42]" 1 h(II' I2) = i + a(I- 3)v~

(3)

where I is the Wagner's generalized invariant [36], written as" I = ~I 1 + (1 - ~) I2

(4)

with" 11 = I2 = 72 + 3

in shear

(5)

11 = exp(2e) + 2 exp(-e) and 12 = exp(-2s + 2 exp(s

in elongation

(6)

P a r a m e t e r s a and ~ are relevant to the shear and extensional behaviour, respectively, while parameter b acts in both flow situations.

292 The adjustable parameters of equation 3 are listed in Table 2. To get an appropriate set of values, a minimization procedure was carried out, where both shear and extensional data were used. Table 2 Parameters of the damping function for LDPE and LLDPE. Parameters

a

~

b

LLDPE

0.086

0.02

2.56

LDPE

0.084

0.019

2.06

3~2. Differential constitutive equations: the GOB and m P T r m o d e l s The Cauchy stress tensor ~ can be written as follows : ~ =-p'S

+~v

+ ~

where p' is the isotropic part of the stress tensor, 8 the unit tensor, ~ v

(7) the

viscoelastic part of the extra-stress tensor and ~ the purely viscous part, usually refered to as solvent behaviour. In this paper, the GOB model is defined as follows : ~r = ~r~ + ~r.

(8)

V

~

+~(~)~v =

2 n v ( ~ ) ID

'Jl~i~ = 2 TIN( ~ ) ]]~) The triangular superscript denotes the upper-convected derivative and

(9) (10) (

T ) is the second invariant of the strain rate tensor ID. To simulate the behaviour of the LLDPE melt, we retained the following Carreau-Yasuda relationships for the viscosity and the relaxation time: (i) viscosity:

11( ~ )= 'I1N(~) -t- 'I~v( ~ )= 'I10[1 +( ~0 ~)a] (m'lya

(11)

where m is the flow index (m = 0.19), ~o the zero-shear viscosity (TIo = 12500 Pa.s), ko the characteristic time (~.o= 0.0796 s) and a = 0.7 at 175 ~

293 (ii) relaxation time:

x(

(12)

)=

The parameter p = 0.8 defines the relaxation time dependence upon the strain rate and C = 0.9 s is a time constant. The characteristic time Xr is 0.4 s and the parometer b is 0.48 at 175 ~ The ratio of the solvent viscosity to the total viscosity retained in the model is 1/9, generally used in most of the papers on the subject. For the P2~ model with a single relaxation time, we adopt the general form for the extra-stress tensor: Q

f(Tr(~ v )) ~rv + ~ ~Tv = 2 ~vo ]D

(13)

Q

The time derivative ~ v is the linear combination of the upper and lowerconvected derivative of ~ v , namely the Gordon - Schowalter derivative [43]: v

a

~

~r~ = ( 1 - 2 ) ~Tw+ ~ V

with O < a < 2

(14)

where a is the slip parameter, introduced by Johnson and Sego!man [44]. The function f (Tr ( ~Tv )) has an exponential form, given by Phan Thien [45]:

f(Tr('Tv))

(15)

= exp(~v~)Tr(~ v ))

where E: is a material constant. If the slip parameter a is a non-zero constant, the requirement t h a t the shear stress be a monotonically increasing function of the shear rate in simple shear flow imposes a constraint upon the viscosity ratio. Using a spectr~m of relaxation times loosens this constraint and allows for more realistic fitting of the rheological data. Therefore, using multiple relaxation times, Equation 13 yields"

E XPLT]vi T r ( ~ )

]~

+ ~ i ~ v t = 2Tl~

with

~ v = Z ~rv~ 1

The values of parameters a and E for both samples are listed in Table 3.

(16)

294

Table 3 Parameter a and e of the P2~ model for LLDPE and LDPE. ,,

Parameters

,

,m,

a

LLDPE

0,35

0,06

LDPE

0,15

0,025

10 6 l---t

u}

10 5

c r

04

mZ

o.

10 3

I02 L-10-3

%

10-2

10-I 10 ~ 101 10 2 Shear or Elongational Rate Is - 1 ]

10 3

Figure 1. Steady shear and extensional viscosities and normal stress difference versus strain rate for LLDPE at T=160~ ( o , ~ ) experimental, (.... ) F I ~ model, (--) Wagner model. The comparison of the predictions given by these constitutive equations in steady-state shear and extensional simple flows are summarized in Figs. 1 and 2. The multimode Wagner-type and PTT models fit the experimental data r a t h e r well, that justifies the use of these constitutive equations to simulate the entry and exit flow of LLDPE and LDPE. The GOB model is not represented because of the prediction of infinite extensional viscosities in the long time range. Nevertherless, the parameters of equations 11 and 12 have been determined from the shear experiments shown in Figs. 1 and 2.

295

10 6 r-'-i

tD

,;

l0 s

o o

r

0

w~

c au. 1

04

fflz

~

103

I0 2 10-3

10-2 i0-I I0 0 I01 I0 2 Shear or Elongational Rate [s - I ]

103

Figure 2. S t e a d y shear and extensional viscosities and normal stress difference versus strain r a t e for LDPE at T= 160~ (o,o~) experimental, (.... ) PTT model, (--) Wagner model. 4. F L O W G E O M E T R I E . S A N D E X P E R I M E N T S Two types of flows were investigated. At CEMEF, experiments on a plane geometry were carried out. Birefringence m e a s u r e m e n t s were performed, the results of which are presented in Section III-1. Comparison with computed results will be presented in p a r a g r a p h 6.2. At LRMP, flows in aYisymmetric contractions were studied. We present in the next pages the experimental results obtained for pressure drops, entrance pressure losses and die swell. 4.1. F l o w c u r v e s a n d e n t r a n c e p r e s s u r e l o s s e s

Capillary experiments were performed at 160 ~ with a c o n s t a n t speed capillary rheometer Instron ICR 3211. The axisymmetric converging test section of half-angle 45 ~ is defined as follows : the upstream and downstream radii are 4.76 mm and 0.65 ram, respectively, so t h a t the entrance contraction ratio is 7.32. In addition, the capillary length-to-diameter ratios range from 1 to 20 and the shear rate can be varied from 3 to 150 s -1 (Figs. 3 and 4), so t h a t flow m e a s u r e m e n t s were generally done in the shear-thinning region, as shown in Figs. 1 and 2. It should be pointed out that the m e a s u r e m e n t s were limited at shear rate of 150 s -1 by the onset of melt fracture. In addition, e n t r a n c e p r e s s u r e losses were e s t i m a t e d u s i n g Bagley's procedure. The set of values presented in Figs. 5 and 6 were obtained from flow measurements with two different rheometers, one equipped with a load sensor, the

296 other with a pressure sensor. Scattering of the data is observed, especially for L L D P E for which the average values of entrance pressure losses are given with errors as high as 50%. I0 s

-

EL. EL o k.

@ 0

10 7

a

{D

A

:::} ID

A

E!

L

El

10 6

0

I:1.

0 I--

Apparent Shear Rate [s - I ]

o

IOs

10 ~

9

i

|

|

9 " ' l |

|

|

9

9

9 . . . |

10 a

,L

i

9

.

!

. . . I

10 2

10 3

Figure 3. Total pressure drop versus apparent shear rate for LLDPE at T=160 ~ with various die length-to-diameter ratios: (o) 0.96; (~) 2.88; (A) 4.81; (0) 19.23 lO s

-

Q.

o

L_

107

Q L

O.

I0 6

4-J

Apparent Shear Rate [s - I ] 10 5

I0 ~

. . . . . . . .

101

,

.

.

.

.

.

.

i|

102

t

i

.

,

.

,..!

103

Figure 4. Total pressure drop versus apparent shear rate for LDPE at T=160 ~ with various die length-to-diameter ratios: (O) 0.96; (O) 2.88; (A) 4.81; (0) 19.23

297

10 7 0. In (i 0 LC~

I0 s

A

A

A A A AA ~A A

A

Q)

L. D

0.

A~

10 5

A

0 C L (-

ILl 1 0

4.

_

9

10- I

. . . . . . .

'

I0 o

9

9

9 . . . . .

,

9

101

....

I

.

102

.

.

.

.

.

.

.

.

.

.

,

103

,.

.

, -

....

,

104

Apparent Shear Rate [s - I ]

Figure 5. Entrance pressure losses versus apparent shear rate for LLDPE at T=160 ~

10 7 (I. Q. 0 t_ C~

106

D t~ k. O-

~D

~A iO s

A

o c

L -bJ tD

10 4 10-i

i

9

9 ..,..I

.,

|

|

.,.,,I

.

,

.

.....i

.

.

,

,,...L

I0 ~ I0 i 10 2 10 3 Apparent Shear Rate [s - I ]

.

.

,

I.,.,!

10 4

Figure 6. Entrance pressure losses versus apparent shear rate for LDPE at T=160 ~

298 4.2. E x t r u d a t e s w e l l The polymer melt was extruded at 160~ downwards directly into air. The diameter of the passing extrudate was then measured at 6.5 mm from the die exit, using an electro-optical device ZIMMER OGH. All measurements were made approximately 5 mm back from the lower end so that sagging under gravity could be n e g l e ~ [46]. 1.4--

1.3

a&&&&

"~

-o L_ 4-~

x

L,J

9 1.2

f

0

~o ~ .~ O 20 o

i

&

&&&&&~'

0

A

&

a

0

I.I ~ o

~o 1.O "n~ 0

I

, l

=

l

2

3

.

_1

,

i

4

,

!

5

Z/Ro

,

!

.

7

6

i

8

Figure 7. Evolution of the free sin-face at the die exit; L/d =19.23, ~a= 3.3 s -1 (A) LDPE, (o) LLDPE. 2.2

2.0

AA""A&&a ="AAAA

& & A A &

&&&&AA& A&&A&"~A&&A'&'&'&&A A

1.8 1.6

1.4

t

0

0

o

0

o

o

o

1.2 1.0

0

1

2

3

4

5

6

7

8

9

I0

Z/Ro

Figure 8. Evolution of the free surface at the die exit. L/d = 0.96, ~s= 33 s -1 (A) LDPE, (o) LLDPE.

Ii

12 13

299 In order to get complementary information on the evolution of the shape of the extrudate at the die exit, photographs were taken simultaneously for each flow condition. Examples of free surfaces build establishments are presented in Figs. 7 and 8, respectively for smooth (long dies and low shear rate) and for stronger flow conditions (short die and high shear rate). It is clear that strong flow conditions enhance the differences between LLDPE and LDPE. While LLDPE exhibits rapid evolution towards a constant diameter, instabilities occur for LDPE, which correspond to the onset of melt fracture. In Figs. 9a and 9b, the extrudate swell ratios of the two polyethylenes are plotted as a function of the die length-to-diameter ratio, for various shear rates. The extrudate swell values for a given shear rate are higher for short dies and then decrease to reach asymptotic values as the length-to-diameter ratio increases. This enhanced swelling for short dies and high shear rates as opposed to swelling obtained for long dies and low shear rates is related to the fading memory of the viscoelastic fluid, when its residence time inside the capillary becomes shorter. For long dies (e.g. L/d = 20), the =equilibrium" extrudate swell values of LLDPE and LDPE are not so different, as it can be seen in Fig. 10, but due to the strong viscoelastic behaviour of LDPE, its extrudate swell values for short dies and high shear rates are eonsiclerably higher than those of LLDPE.

1.6

2.2 A

o

CO

1.4

o

~

|

A

D

~

n

Or)

(P

A

~

"O

O

L.

x

Q

1.8

O

I::]

"~

O

L.

1.2

x

td

I,J

1. 0

.

0

.

.

.

.

5

a - LLDPE

III

10

L/d

"

I

15

I

I1

20

0

O

O

A 0 0

1.4

t. 0

--

0

I

I

I

5

10

15

b - LDPE

C/d

--'

20

Figure 9. Extrudate swell versus die length-to-diameter ratio, at various apparent shear rates. (T = 160 ~ (o) 3.3 s-i; (O) 11 s-l; (h) 33 s-l; (0) 110 s -1

300

2.22.0

OO

"0 t_ .&a x bU

1.8 1.6

9

1.4

g

8

o

o 0

1.2

o

~n

........

I0 ~

t

,

I0 t

........

,

102

........

,

103

Shear Rate Is - I ]

Figure 10. Extrudate swell of LLDPE and LDPE versus apparent shear rate at T = 160~ Open symbols- L/d = 19.23; filled symbols" L/d = 0.96. (O,o) LLDPE; (A,A) LDPE 5. N U M E R I C A L M O D E L S 5.1. T h e s t r e a m . t u b e m e t h o d 5.1.1. Introduction Major advances in numerical simulation of complex flows of viscoelastic fluids were particularly significant in the recent years for differential constitutive equations. For fluids with memory, more theoretical and numerical difficulties arise. The approximation of kinematic tensors leads one to consider the particle's kinematic history along a streemline, the points of which do not pass in general through the mesh nodes of the physical flow domain, in the context of classic finite-difference or finite-element techniques. Different approaches were proposed in the literature, in two-dimensional flow situations [21,22,47,48]. The fact that, to our knowledge, few numerical simulations [49] of memory-integral viscoelastic 3D flows have been attempted up to the present time is due mainly to significant problems in defining accurate parametric equations for warping curves. A different technique for computing the flow of memory-integral fluids was proposed by Papanastasiou et al. [20], where a Protean coordinate system introduced by Duda and Vrentas [50] and developed by Adachi [51-52], was used. In the Protean system, one coordinate is the stream function. The stream-tube method is more closely related to the Protean coordinate approach. It refers to the flow analysis introduced some years ago by Clermont [40,53], which may be applied to the study of two- or three-dimensional duct or free surface flows [54-56] and pure circulatory or vortex flows [57]. In this analysis, the unknowns of the problem are, in addition to the pressure p, a one-toone transformation between the physical flow domain D (or a subdomain D* of D)

301 and a transformed domain D', used as computational domain, where the mapped streamlines are parallel and straight. Although this analysis means considering only open streamlines, main flows in ducts involving recirculations can still be calculated with the stream-tube analysis, as was done in recent numerical works [26,55]. The stream-tube method is presented here in the scope of numerical calculation of flows of fluids obeying viscoelastic (differential or integral) constitutive equations. 5.1.2. Stream-tube method in two-dimensional flows The main features of the stream-tube method in 2D and 3D flows, discussed more extensively elsewhere [40, 53, 55], are now summarized for two-dimensional situations. Flows with open streamlines are considered. The main flow region D* of the physical domain D (D ~ D*) is mapped into a domain D' such t h a t the transformed streamlines are straight and parallel to an assumed m a i n direction Oz of the flow. An example corresponding to 2D flows is illustrated in Fig 11.

~ PHYSICAL (r, z)

I

L

Stream tube

.........

zo

' -

I

; !

y

, I ! I

k MAPPED DOMAIN D'

'

t

......"..... "'" \ "\ ............ "~, " ' j Stream band

(R, Z)

; _ I I

7-o

,, i . i ~ ~ 7 / , . / / i

.-1 1 1 J ,.111 i~,~.,1 1 J 1111..-+~i~

Figure 11. Physical and mapped domains in a two-dimensional problem In two-dimensional flows, the domains D' and D are related by a transformation '% by means of a mapping function f , a priori unknown, such that" x=f(X,Z),

y=Y, z=Z,

(17)

with the following upstream boundary condition" x = f ( X , Zo)=X;

y = Y ; z=Zo.

(18)

These equations refer to cartesian coordinates Xi (X 1-- X; X 2 =y, x 3 = z) and XJ (Xl= X; X2 =Y, X3 = Z) in domains D and D', respectively. The mapped domain D' is a cylinder of generators parallel to OZ, of basis identical to the cross section So of the physical domain D at section Z0= z o. The following assumptions are made :

302 (i) At sections z~

a)

i

o

8: o

o

0,05 0,025

-0,02

' -20

-10

0

Axial

O,

0

o

0

0,02

0

9

~

e

De

w 0 L_

20

(ram)

b)

A m

v

distance

10

0

.

.

.

d

.

~

9

"o o ~

eb~176176 Q o

a

.

| m qmo X W

-20

-10 Axial

0 distance

10

20

(ram)

Figure 22. Viscoelastic extra-stress components (a: Tvxx, b" Tvyy) along the symmetry axis. (O) generalized Oldroyd-B model ; (0)multimode Phan-Thien Tanner model.

6. C O M P A R I S O N B E T W E E N N U M E R I C A L R E S U L T S A N D EXPERIMEN2~

Flow experiments were carried out at LRMP for axisymmetric contractions and at CEMEF for plane geometries. Numerical simulations were performed at Laboratoire de Rh~ologie, with Wagner memory-integral constitutive equations, with the stream-tube method (sub-section 5.1) and at CEMEF, w h e r e t h e finite-

318

element method was combined with the use of differential constitutive equations (GOB and mPTT models). At Laboratoire de Rh6ologie, different procedures were used for the converging flows and extrudate swell: (i) For converging flows, the peripheral stream-tube was considered, with modified Hermite elements (Sub-section 5.1). The solution of the equations was performed with a Trust-Region optimization algorithm. The use of a stream-tube analysis associated with this resolution algorithm did not require, for a given flow rate, a prior solution at a close value of the flow rate under consideration. The code was implemented on an Apollo 425 work station and no limitation based on the flow rate was found in the flow calculations. The rectangular mesh involved 24 to 36 elements. The lengths Lu and Ld upstream and downstream of the contraction section were taken as L u = 13 r 1 , 30 r 1 < Ld~ 45 r 1 (rl denotes the downstream tube radius), in order to obtain relaxation of stresses given by the memory-integral equations. For all runs, the number of iterations was found to be lower than 25. Given a flow rate, the CPU time required for convergence was found to be of order of 8 hours. (ii) For the extrudate swell problem, the code was implemented on a HP 9000 work station.The peripheral stream-tube involved a computational domain in the tube and the jet domains such that ( z 0 - Zp)/r 1 = 5 ; ( z 2 - Zo)/r 1 = 10 (Subsection 5.1). The computational algorithm used for solving the equations was the Levenberg-Marquardt optimization algorithm [67]. Given a flow rate, the CPU time for obtaining a swell surface was found to be of order of 1 hour. No limitation of convergence was found for the flow rates investigated.

At CEMEF, the numerical results were obtained by using a generalized Oldroyd-B model and a multimode Phan-Thien Tanner model, with a LesaintR a v i a r t quasi-Newton finite element formulation (Sub-section 5.2). The computations were run on an IBM Risk 6000/580 work station. Convergence was obtained for most of the experimental flow situations presented in section 4, depending on the flow rates and on number of relaxation times accounted for in the mPTT model. The CPU time varies from 10 to 70 hours, but the storage requirements remained limited. It should be pointed out that the method has provided velocity and stress fields for all types of 2D planar and axisymmetric geometries investigated experimentally (contraction and converging flows for short and long dies, extrudate swell).

319

6.1. Axisymmetric flows 6.1.1. Flow curves and entrance pressure losses for L L D P E and L D P E Figure 23 compares, for various L/d ratios,the experimental pressure drop and the numerical radial stress difference along the wall (between 10 m m upstream from the contraction and die exit),as a function of the apparent shear rate (in the reservoir, the wall radial stress C~rris quite equivalent to the m e a n stress in the flow direction,Dzz,which is the so-calledpressure drop measured). For the longest die, the G O B model as well as the multimode P 2 ~ and the W A G N E R models give similar results,in close agreement with experimental data on L L D P E and L D P E (within 15%). Numerical results using the G O B and the m P T T models were also available for short dies.Here again, the agreement is generally good.

IOs a. o.

o L_ C~

3 10 7

L. Q~ 01 L

Q.

1 106

4-'

10 5

I0 ~

..

,

.

.

.

.

.

,I

101

.

9

9

|

|

9 . . l l

10 z

i

l

.

,

,

. , , I

10 3

Apparent Shear Rate Is - I ]

Figure 23. Total p r e s s u r e drop as a function of the s h e a r r a t e at the wall of the d o w n s t r e a m channel. Curve 1 : experimental data for LLDPE (L/d = 0.96). Curve 2 : experimental data for LLDPE (L/d = 4.81). Curve 3 : experimental data for LLDPE (L/d = 19.23). Curve 4 : experimental data for LDPE (L/d = 19.23). (A) : GOB model; (o) : mPTT model; (O, 0) : mWagner model.

Fig 24 compares experimental entrance pressure losses of L L D P E obtained using Bagley correction and numerical results obtained with the three models, b u t using different methods.

320 - When considering the difference between the %r component of the stress tensor at the wall just upstream and just downstream the conical convergent, one may observe t h a t the computed art loss for GOB and mPTT models is far smaller than the experimental one (a factor 5) ; a similar observation has recently been indicated in the work reported by Feigl and 0tfinger [28]. - To the contrary, the performance of the numerical computation using stream-tube analysis combined with a multimode Wagner-type constitutive equation is closer to the experimental results, especially for LLDPE. Unfortunately, quantitative disagreement still remains for LDPE, as shown in Fig. 25. - In numerical simulations involving the GOB or the ~ models, we rebuilt a numerical Bagley correction, deduced from the drop of the %r component, for the three capillary lengths at the same flow rate. We observe that this numerical entrance correction remains smaller than the experimental one, but of the same order of magnitude (30 % difference). Similar results were presented recently by Barakos and Mitsoulis [25].

10 7

-

(g

/

ffl 0

o L

106

i

o

L.

D U~ U) L

o.

~

i0 s

A

U e-

G 6

$-

4.J eLd lO

4

lOo

9

.

.

.

.

. . . I

9

.

.

.

.

. . . I

|

101 102 Apparent Shear Rate [s - I ]

,

.

.

.

.

* * I

103

Figure 24. Entrance pressure loss of LLDPE as a function of the shear rate in the downstream channel. (--) Experimental curve from Bagley correction. Open symbols: numerical computation of A~rr; Filled symbols: numerical Bagley corrections; (O) :mWagner model (~3) : mPTT model; (A) : C~B model.

321

10 7 r--i

In Q. 0 I..

s

(D

t..

Z3

~)

ly)

10 6

0

I,,.

12. 0 eL 4~ eLd

10 5 10 ~

L

.

|

I

. . . I

101

,

i

I

9

9

.

| . . I

i

102

Apparent Shear Rate [s -1]

,I

I

9

9

9 i

,I

103

Figure 25. Entrance pressure loss of LDPE as a function of the a p p a r e n t shear rate in the downstream channel. (--) Experimental curve from BAGLEY correction. (o) numerical computation of AOrr (mWagner model).

6.1.2. Extrudate swell Computations were performed for the long dies using the three constitutive equations. For short dies, only differential models (GOB and reFIT) were tested. 6.1.2.1. Long dies (L/d = 19.23) At low shear rate (3.3 s l ) , experimental and computed free surfaces are compared in Fig. 26. The final extrudate swell values are equivalent, b u t the experimental shape of the free surface is different from the computed ones. In addition, a slight overswelling at the die exit is observed for the GOB model, which is observed neither experimentally, nor using a classic Oldroyd-B model. Another example with the Wagner model at a higher shear rate (~ = 33s "1) is presented in Fig. 27, for LDPE and LLDPE. For both materials, the final extrudate swell values are well-predicted. However, the computed radius close to the die exit increases faster than the experimental one and a constant value is reached in a shorter time than in the real case, especially with LDPE.

322

1.3

1.2

+Z

/

o

o

o

Q

-----

L,'f:

,,, l:

1.0 .o 0 1

2

3

4

5

6

7

8

Z/Ro

9

10

11 12 13

Figure 26. Free surface computation for LLDPE; L/d = 19.23 ; ~ = 3.3 s "1. (0)experimental m e a s u r e m e n t s ; (.... ) GOB model; (--) ~ model.

1+F

0

0

0

0

~O 0 ,0

~176176176

1.3 A

3~ CO

"I3

0

~+

A

A

1.2

L_ 4J x I,I

1.1

1.0 0

1

2

3

4

5

6 7 Z/Ro

8

9

I0

II

12 13

Figure 27. Free surface computation for LDPE and LLDPE with stream-tube analysis and m Wagner model. (a) LLDPE; (o) LDPE; (--) computation; L/d = 19.23, ~ = 33 s "1. All the models point to a n increase in the extrudate swell value with the a p p a r e n t s h e a r rate. The m P T T model u n d e r e s t i m a t e s this effect, w h e r e a s the GOB model overestimates it, especially in the high shear rate range (Fig. 28).

323

1.6

"

1.5 Or}

"0 L

4-J X Ld

1.4

1.3

1.2

1.1

Apparent Shear Rate [s-I] .

-o

.

,

- -

.

,

20

I0

9

,

30

40

Figure 28. Influence of the shear rate on the extrudate swell of LLDPE. (--) experimental ; (o), GOB model ; (e) PTT model. Wagner computed values are in better agreement with experimental results, even for LDPE, provided the apparent shear rate is higher than 20 sec -1. At low shear rates, the predicted extrudate swell ratio is largely underestimated, for both LDPE and LLDPE. (Fig. 29) 1.61 0

(D

~

./..~~--'o"

1.4

0

o

~ o

9

"U Z)

l,-

4-J X LtJ

1.2

0

Apparent Shear Rate [s-I] 1.0

0

0

,

I

20

..,

~

40

9

I

60

,

!

80

,

I

100

.

~

120

.

_. i

140

9

J

160

Figure 29. Extrudate swell versus apparent shear rate.

(---) Experimental data; computed values for LLDPE (O) and LDPE (e).

324 6.1.2.2. short dies Fig. 30 presents the comparison of free surfaces, for a s h o r t capillary (L/d = 0.96) for a strain rate of 33 s "1. In this case, the computed final jet radius value given by the GOB model is very close to the experimental one, while the computed m P 2 ~ value underestimates experiments (about 10 %). Nevertheless, the computed shape of the free surface is very far from the experimental one for the GOB model (a large overswelling at the die exit is observed) as it seems to be homothetic when considering the ~ model.

1.5

1.4

Z

1I I / - " " " -

o

o

~ - - "*

0"

-*

t.2

w

1.1 10

0

I

2

3

4

5

6

7

Z/Ro

8

9

I0

ii

12 13

Figure 30. Free surface computation for LLDPE; IJD = 0.96 ; ~ = 33 s -1. (0): experimental measurements (.... ) :GOB model (--) : mPTT model. The influence of the L/d ratio on the equilibrium extrudate swell is presented in Fig. 31, for three values of shear rate (3.3 s -1, 11 s -1 and 33 sl). The GOB model seems to be insensitive to the L/D ratio at low shear rate and gives a tendency opposite to the experiments at higher shear rate. On the contrary, the mPTT model captures the decrease of the extrudate swell ratio with increasing die lengthto-diameter ratio, but underestimates the values at high shear rates, mainly for the shortest dies.

325

1.6-

(a)

1,5 1.4 o ~

o

4-D

"1o L 4-m X

1.3

w

A

1.2 1.1 0

A

A

I

I

I

I

5

10

15

2O

L/d 1.6-

Co)

1.5 u)

1.4

1.3 w

1.2 1.1

0

I

5

I

I0

L/d

!

I

15

20

Figure 31. Influence of the L/d ratio on the extrudate swell ratio of LLDPE. (a), GOB model ; (b) F ] ~ model. (--) experimental ; (A) ~ = 3.3

S "1 ; ((:3) ~ =

11 s -1 ; (O) ~ = 33

S "1

Fig. 32, which gathers together all the computations and the corresponding experimental results for the long die (L/D = 19.23), confirms that, generally speaking, a better description of extrudate swell experiments is obtained with the multimode PTT and Wagner models instead of the GOB model. For the latter, the agreement becomes poorer and poorer when the shear rate increases. On the

326

contrary, the stream-tube analysis using the Wagner model provides satisfactory agreement with experimental data particularlyin the high shear rate range. 1.6-

0

r

1.4

0

~

C 1.2 0

1.0

1.o

.

rn

,

!

.

1.2 1.4 Experimental Die Swell

.

.

1.e

Figure 32. Comparison between theoretical and experimental extrudate swell.(o) GOB model; (e) F I ~ model; (0,c})Wagner model.

6.2. P l a n a r flows In this Section, the numerical results compared to experimental data were obtained at CEMEF, with the differential constitutive equations already presented. 6.2.1. Linear low density polyethylene Experimental birefringence patterns at 205 ~ and numerical simulation with a GOB model and a mPTT model are compared in Fig. 33. The apparent wall shear rate related to the downstream flow is 25 s "1 and the optical stress coefficient was selected to be 2.1 10-9 m2/N (see Section III-1). Both computations provide a good qualitative description of the birefringence patterns, but the ~ model captures more accurately the shape of the fringes in the downstream region. A more precise comparison can be made by looking at the birefringence changes along the flow axis (Fig. 34). The birefringence increase in the reservoir is similar for both models and very close to experimental values, as already observed with results from other numerical computations [27, 30, 65]. The relaxation along the die land is very consistent for the mPTT model, where the stresses do not relax totally before the channel exit. Results with the GOB model indicate a less realistic relaxation, with a local overshoot near the exit, which is not observed experimentally.

327

Figure 33. Experimental and computed birefringence patterns. (a) GOB model ; (b) ~

model ;~/= 25 s -1.

25 A

to

o

20-

)r

0 c

9 15

0

o~ 10_ c

m L.

L_

~

5~'ql~mqb ~ ~

!

-20

!

!

-10

0 Axial

distance

10

20

(mm)

Figure 34. Birefringence along the flow axis (T = 205 ~

~ = 25 s-l).

(0) experimental measurements ; ( ~ ) GOB model ; (- - -) mPTT model.

30

328 At higher shear rates, similar results are obtained (Figs. 35 and 36). We observe again that a total relaxation is never obtained as in the experiment, where the fringe of order zero is never visible in these flow conditions.

Fig.35 " Experimental and computed birefringence patterns; ~ = 40 s'l. (a) GOB model ; (b) ~ model. tO

o30 x ~p

o20 (P C L. N'='

~10 L.

co 0 -20

-10

0

Axial distance

10

20

(mm)


~f = 40 sl).

(e) experimental measurements ; ( - - ) GOB mode] ; (- - -) m P 2 ~ mode].

30

329 The flow computations in a short die (here T= 145 ~ obviously imply free surface determination corresponding to the swell phenomenon. It may be seen in Fig. 37 that the general birefringence patterns are similar, even if the results of the mPTT model seem more realistic in the downstream region than those of the GOB model, for which discontinuities appear along the centreline, apparently due to an incorrect determination of the relaxation time at very low shear rate.

Figure 37. Flow birefringence patterns : T = 145 ~

? = 35 s -1.

(a) P2~ model (b) experiments (c) GOB model.

330

60 A

o T-

x

40-

_

o o c Q c t...

_

;e

%0

20-

:I

\e

Q

~

k. =u

m

0 -20

- -

-15

I - -

|

-10

|

-5

0

Axial distance

5

10

15

20

(mm)


7 = 35 s'l).

(O) experimental measurements ( - - ) GOB model (- - -) ~ model This is confirmed by the evolution along the centreline (Fig. 38), where the relaxation of the GOB model appears to be too slow, whereas the mPTT results are in very nice agreement with the experiment. 6.2.2. Low density polyethylene. LDPE is a highly branched material, whose flow behaviour exhibits some peculiarities, which constitute a good test for a numerical simulation. As presented in Section III-1, birefringence patterns are perturbed a r o u n d the r e - e n t r a n t corner, which leads to the appearance of W-shaped fringes at the entry of the die land. It can be seen in Fig. 39 that the mPTT model allows on to obtain these characteristic shapes (the computation is related to experiments carried out at 175 ~ and 21 s -1, for an estimated value of the stress optical coefficient of

331

1.9 10 -9 m2/N). By comparing with Fig. 33, obtained under similar conditions for a LLDPE melt, the difference is really significant. If we look at the principal stress difference profile ~ - o~I within the die gap in the downstream section (Fig. 40), we observe for the LDPE a non-monotonic evolution near the land entry, which tends towards a more regular shape when moving downstream. For LLDPE, the corresponding profiles are totally monotonic and only change slightly along the flow. Conversely, the evolution of birefringence along the centreline is very regular and shows a slow mechanism of relaxation, which is also very well captured by the model (Fig. 41). From these results, it may be pointed out that the mPTT model is able to predict differences in flow behaviour between linear and branched polymers, which proves both the quality of the computation and the efficiency of this constitutive equation.

I

Figure 39. Flow birefringence pattern ( ~

model, T = 175 ~

~ = 21 sl).

332

~

0,25

.~ymmetry

line

(a)

dle

wall

0,2-~ A

r

a.

:S

0,15

m B

u

0,1

| m

u

0,05 I

0

I

0,5

1

Flow depth (ram)

0,25

symmetry

line

(b)

die

wall

0,2 A

r

n_ v

0,15

m n

t:) | m

0,1

t:)

0,05 0

0

0,5 Flow depth

1 (mm)

Figure 40. Computed profile of principal stress difference across the die gap a) L D P E

b) LLDPE 9 ( o ) - 2 m m d o w n s t r e a m of the section of contraction (e) 98 m m d o w n s t r e a m of the section of contraction

333

A

3O

>r v

2O o c c

":"

10 9I

I,., m

I

~ ~ O

m

0 -15

-10

-5

0 Axial

5

10

distance

15

20

25

30

(mm)


q = 21 s-l).

(o) experimental measurements ; ( - - ) mP2~ model.

7. CONCLUSIONS This paper reports experiments and numerical simulations related to a linear low-density polyethylene (LLDPE) and a low-density polyethylene (LDPE), in a significant number of axisymmetric and planar mixed flows. Converging and abrupt contraction geometries involving short and long dies were considered as well as extrudate swell flows occurring at the exit of the ducts under investigation. The rheological behaviour of the two polymers was determined using classical techniques of rheometry, already described in Chapter II.1 (rotational and capillary rheometers for shear viscosity and first normal stress difference measurements; Cogswell method for the elongational viscosity). Different viscoelastic constitutive equations were adopted for modelling the experimental data of both fluids: - a memory-integral K-BKZ constitutive equation, using a damping function of the Wagner type depending on the invariants of the Finger strain tensor; - t w o differential constitutive equations : a generalized Oldroyd-B model (GOB), with viscosities and relaxation times depending on the second invariant of the rate-of-strain tensor, and a multimode Phan-Thien Tanner model, with a spectrum of relaxation times and viscosities and two adjustable parameters. Two numerical methods, very different in essence, were carried out to simulate the flows : - a stream-tube method, - a finite element method.

334 Since significant comparisons were made between experimental results and those from both numerical techniques, no global comparison exists between all the experimental measurements, the constitutive equations and the two numerical methods. For example, the stream-tube method has only been tested with a Wagner memory-integral equation, and the finite element method only with differential GOB or mPTT equations. The Wagner equation has been investigated to simulate flows in axisymmetric geometries and for long dies for a wide range of flow rates. The differential constitutive equations have been tested for all kinds of geometries, but concerned a narrower range of flow rates. Nevertheless, the work reported in this paper provides systematic tests on different constitutive equations, with the same polymers and over a wide range of flow conditions. In planar flow situations, the qualitative agreement between computed stress fields and experimental flow birefringence patterns is to be underlined. Concerning the constitutive differential equations used to model the polymers, the model provides a better quantitative agreement, especially along the flow axis, than the GOB model. The mPTT model is able to capture the differences in stress patterns which are observed between LLDPE and LDPE. We think that this difference is mainly due to a poorer description of the relaxation times of the polymers in the GOB model. In axisymmetric flow situations, the global pressure drop in a capillary rheometer is well described by the three constitutive equations. If one focuses on the entrance pressure drop, the numerical entrance pressure drop related to Bagley correction is found to be less important than the corresponding experimental data for the differential models for LDPE and LLDPE melts. For the Wagner integral constitutive equation, the computed entrance pressure drops are found to be lower for both fluids, but the computed values are closer to the experimental data for LLDPE than those related to the LDPE melt. This descrepancy, previously reported in the literature, needs fugher investigation. When considering the extrudate swell, none of the three constitutive equations provides a good estimation in all processing conditions. The integral Wagner model gives good predictions at high flow rates, but underestimates extrudate swell at low flow rates. Concerning the differential models, a good agreement is obtained at low flow rates, but numerical convergence remains difficult at higher flow rates. The influence of the L/D ratio is opposite to experimental observations for the GOB model and more realistic for the mPTT model. As a consequence, without consideration of numerical schemes defined for solving the equations, the extrudate swell flow appears to be one of the more discriminating experiments for testing constitutive equations.

335

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

D.V. Boger, Ann. Rev. Fluids Mech. 19 (1987) 157. S.A. White, A,D. Gotsis and D.G. Baird, J. Non Newt. Fluid Mech. 24 (1987) 121. G.H. Mc Kinley, W.P. Raiford, R.b, Brown and R.C. Armstrong, J. Fluid Mech. 223 (1991) 411. F.N. Cogswell, Polym. Eng. Sci. 12 (1972) 64. J.L. White and A. Kondo, J. Non Newt. Fluid Mech. 3 (1977) 41. S.A_White and D.G. Baird, J. Non Newt. Fluids Mech. 20 (1986) 93. X.L. Luo and E. Mitsoulis, J. Rheol. 34 (1990) 309. H. Miinstedt, J. Rheol. 24 (1980) 847. H. Miinstedt and H.M. Laun, Rheol. Acta 20 (1981) 211. D.V. Boger, Pure Appl. Chem. 57 (1985) 921. P.J. Coates, R.C. Armstrong and A. Brown, J. Non Newt. Fluid Mech. 42 (1992) 141. R. Keunings, J. Non Newt. Fluid Mech. 20 (1986) 209. J.M. Marchal and M.J. Crochet, J. Non Newt. Fluid Mech. 26 (1987) 77. R.C. King, M.R. Apelian, R.C. Armstrong and R.A, Brown, J. Non Newt. Fluids Mech. 29 (1988) 147. D. Rajagopalan, R.A. Brown and R.C. Armstrong, J. Non Newt. Fluids Mech. 36 (1990) 159. F. Debae, V. Legat and M.J. Crochet, J. Rheol. 38 (1994) 421. B. Berstein, K.&Feigl and E.T. Olsen, J. Rheol. 38 (1994) 53. R. Keunings and M.J. Crochet, J. Non Newt. Fluid Mech. 14 (1984) 279. A.C.Papanastasiou, L.E. Scriven and C.W. Macosko, J. Rheol. 27 (1983) 387. A.C. Papanastasiou, L.E. Scriven and C.W. Macosko, J. Non Newt. Fluid Mech. 22 (1987) 271. S. Dupont and M.J. Crochet, J. Non Newt. Fluids Mech. 29 (1988) 81. X.L. Luo and R.I. Tanner, Int. J. Num. Meth. in Eng. 25 (1988) 9. A. Goublomme, B. Draily and M.J. Crochet, J. Non Newt. Fluids Mech. 44 (1992) 171. A. Goublomme and M.J. Crochet, J. Non Newt. Fluids Mech. (1993). G. Barakos and E. Mitsoulis, J. Rheol. 39 (1995) 193. Y. Bdreaux and J.R. Clermont, Int. J. Num. Meth. Fluids Vol 21, 5 (1995) 371. D.G. Kiriakidis, H.J. Park, E. Mitsoulis, B. Vergnes and J.F. Agassant, J. Non Newt. Fluid Mech. 45 (1992) 63. K.A. Feigl and H.C. ()ttinger, J. Rheol. 38 (1994) 847. B. Knobel, Ph.D. Thesis, Diss. Zurich Eth N ~ 9480 (1991) H. Maders, B. Vergnes, Y. Demay and J.F. Agassant, J. Non Newt. Fluid Mech., 47 (1992) 339. H.J. Park, D.G. Kiriakidis, E. Mitsoulis and K.J. Lee, J. Rheol. 36 (1992) 1563. R. Ahmed, R.F. Liang and M.R. Mackley, J. Non Newt. Fluid Mech. 59, (1995) 29. M. Van Gurp, C.J. Breukink, R.J.W.M. Sniekers and P.P. Tas, Spie, 2052 (1993).

336 34. E. Mitsoulis and H.J. Park, Theor. and Appl. P~eol. Proc. XIth Int. Cong. on Rheology, Brussels (1992). 385. 35. D.G. Kiriaskidis and E. Mitsoulis, Adv. in Polym. Tech. 12 (1993) 107. 36. M.H. Wagner, Rheol. Acta. 18 (1979) 681. 37. J. Meissner, Pure Appl. Chem., 42 (1975) 551. 38. C. Beraudo, T. Coupez, B. Vergnes, C. Peiti and J.F. Agassant, Les Cahiers de Rh~ologie, XI, 3-4 (1993) 303. 39. C. B~raudo, Th~se de Doctorat (1995) 40. J.R. Clermont, C. R. Acad. Sci. Paris, s~rie II-1 (1983) 297. 41. C. Carrot, J. GuiUet, J.F. May and J.P. Puaux, Makrom. Chem. Theory Simul. 1 (1992) 215. 42. P.L. Soskey and H.H. Winter, J. Rheol. 2 (1984) 625. 43. R.J. Gordon and W.R. Schowalter, Trans. Soc. Rheol. 16 (1972) 79. 44. M.W. Johson and D. Segalman, J. Non Newt. Fluids Mech. 2 (1977) 255. 45. N. Phan Tien, J. Rheol. 22 (1978) 259. 46. J. Guillet and M. Seriai, Rheol. Acta. 30 (1991) 540. 47. M. Viriyayuthakorn and B. Caswell, J. Non-Newtonian Fluid Mech, 6 (1980) 245. 48 X.L. Luo and E. Mitsoulis, J. Num. Meth. Fluids 11 (1990) 1015. 49 K.R. Rajagopal, Theor. Comput. Fluid Dynamics, 3 (1992) 185. 50. J.L. Duda and J.S. Vrentas, 22 (1967) 855. 51. K. Adachi, Rheol. Acta, 22 (1983) 326. 52. K. Adachi, Rheol. Acta, 25 (1986) 555. 53. J.R. Clermont, Rheol Acta 27 (1988) 357. 54. P. Andr4 and J.R. Clermont, J. Non-Newtonian Fluid Mech. 39 (1990) 1. 55. J.R. Clermont and M.E. de la Lande, Theoret. Comput. Fluid Dynamics, 4 (1993) 129. 56. J.R. Clermont and M.E. de la Lande, J. Non-Newtonian Fluid Mech. 46 (1993) 89. 57. J.R. Clermont, M.E. de la Lande, T. Pham Dinh and/~ Yassine, Int. J. Num. Meth. Fluids 13 (1991) 371. 58. M. Normandin and J.R. Clermont, J. Non-Newtonian Fluid Mech. 50 (1993) 193. 59. Y. B~reaux, Rapport de DEA, Universit~ de Grenoble (1992). 60. J.R. Clermont, Rheol Acta 32 (1993) 823. 61. A. Yassine, Th~se de Doctorat de Math~matiques, Universit~ de Grenoble, 1989. 62. P. Lesaint, P.A. Raviart and C. de Boor Ed., Academic Press, (1974) p. 89. 63. M. Crouzeix and P.A. Raviart, Rairo, A3, 3 (1973) 33. 64. M. Fortin and A. Fortin, Commun. Appl. Num. Methods, 1 (1985) 205. 65. P. Beauffls, B. Vergnes and J.F. Agassant, Int. Po, Report 189 (1971) 66. J. Batchelor and F. Horsfall, Rubber Plast. Res. Assoc. (1971) 67. A Gourdin and M. Bouhmarat, M~thodes Num~riques Appliqu~es, Tec et Doc Ed. Lavoisier, Paris (1989).


337

Slip at t h e w a l l L. L6ger, H. Hervet and G. Massey

Laboratoire de Physique de la Mati6re Condens6e, URA C.N.R.S. 792, Coll6ge de France, 11 Place Marcelin - Berthelot, 75231 PARIS Cedex 05, FRANCE 1.

INTRODUCTION:

It has been suspected for a long time that flows of high molecular weight polymers couJ exhibit a non-zero boundary condition for the velocity at the wall, contrary to what is usual J simple liquids. If such is the case, this is of great practical importance, as for example in extrusion process, the shear rate and the stress experienced by the polymer molecules at tt interface will condition the properties of the extrudate. A number of investigations have tht attempted to characterise wall slip in polymer systems, first through rheological macroscop: characteristics such as pressure drop as a function of flow rate, for various flow geometries [i 8], then trying to characterise the flow behaviour as a function of the thickness of the liquJ [9,10], or determining the velocity gradient by the use of tracer particles or by velocimeu [ 11,12], and more recently, measuring directly the local velocity in the immediate vicinity the wall (within 1000A from the wall) [ 13,14,15]. At the same time, and because instability J an extrusion process and extrudate defects have been related to the onset of flow with sli boundary conditions [2,16,17], several attempts have been made to model flows with slip, an to relate the onset for strong slip with the molecular characteristics of the polymer [ 18 - 25]. The macroscopic investigations show that the appearance of wall slip depends on tt system polymer/surface under investigation. For some systems, and some geometries of flo~ wall slip seems to always be present, whatever the stress level, in the range accessible in tk macroscopic experiments [2] (but the data of reference [2] may have been misinterpreted, entry corrections, at capillary heads, have not been introduced), while in many other cases limiting shear rate has to be reached before slip can be macroscopically detected [26,4,3,12]. detailed analysis of the onset of slip is not however easy to conduct from macroscopi experiments: if the extrapolation length of the velocity profile to zero remains small compare with the thickness of the flowing liquid, the resulting change in the relation pressure drc versus flow rate may be hardly detectable. The direct investigation of the velocity at the wal] thus appears as a key step in the understanding of the flow behaviour of molten polymers. We present here an investigation on the role of polymer/surface interactions on tk existence of wall slip. Direct measurements of the local velocity at the wall allow one t investigate how wall slip is influenced by polymer chains anchored to the surface, b adsorption or grafting [14,15]. Such local measurements are limited to a small number polymers, due to the technique used, but they allow unambiguous comparisons with tk molecular models which have been recently developed and can thus be validated. The~, experiments and models demonstrate that for surfaces with a weak roughness, the ke parameters which govern the existence of a shear rate threshold for the onset of strong slip~ the wall are the deformability of surface anchored polymer chains and their degree interdigitation with the bulk liquid. Several regimes of wall slip can thus be identified. The should, of course, also show up in macroscopic rheological measurements, each regirr corresponding to a particular friction law. The understanding of how the polymer surfac

338

interactions govern the structure of the surface anchored chains and as a consequence the slip regime should open the way to the design of tailored surfaces adjusted for a particular application: efficient extrusion, controlled friction .... 2. L O C A L D E T E R M I N A T I O N OF T H E V E L O C I T Y AT T H E W A L L :

2.1 Measuring techniqueUnambiguous determination of the conditions under which slippage occurs requires a technique able to measure the velocity of the fluid in the immediate vicinity of the solid wall over a thickness comparable to the size of a polymer chain, i.e. a few tens of nanometers. Classical laser Doppler velocimetry does not meet this requirement even if it allows for the determination of velocity profiles which clearly reveal a non-zero velocity within typically a few 10 ~m from the wall. We have developed a new optical technique, Near Field Velocimetry (N.F.V.) [14], which combines Evanescent Wave Induced Fluorescence (E.WF.) [27] and Fringe Pattern Fluorescence Recovery After Photobleaching (F.P.F.R.A.P.) [28]. The former technique gives the spatial resolution normal to the solid wall, while the latter one enables the determination of the local velocity of the fluid. A major constraint of the technique is that it needs polymer molecules labelled with an easily photobleachable fluorescent probe. The sample cell is schematically presented in Fig. 1. A drop of fluorescently labelled polymer melt (refractive index n2) is sandwiched between two plane silica surfaces (refractive index n2 < n l) held at a distance d (d = 8 ~tm in all the experiments presented here) by two mylar spacers. Care is taken to maintain the two limiting surfaces parallel when adjusting the mechanical clamps which hold them at the distance d, checking for no or only one or two visible interference fringes in white light between the beams partially reflected by the two surfaces. In order to avoid parasitic hydrodynamic effects, the drop is never in contact with the mylar spacers nor with the edges of the silica plates, as schematically presented on the top view of the cell in Fig. 1b. The upper surface can be translated with respect to the bottom one along a

!ar spacer silicaplate

Vt

top face of the pris PDMS

PDMS n2 s

~

laser beams

I silica pri

.~r/,vnt!

, i

!

top la

lb

Figure 1. Schematic representation of the sample cell, a) Side view of the cell, b) Top view of the cell.

339

direction perpendicular to the fringes, at a controlled velocity, Vt, imposing a simple shear flow to the drop of liquid. The apparent shear rate 4/~pp= Vt/~dd can be varied in the range 0.01s -1 ___?'app -< 40s-1. As flow tracers we use polymer molecules which are chemically labelled with one fluorescent molecule at both ends. Typically one per cent by weight of the total polymer molecules are labelled, maintaining the concentration of fluorescent probes below 10 ppm for the high molecular weights used. To get a flow tracer, one needs to create in the sample a non- uniform concentration of fluorescent molecules, and then, to follow the deformation of this non-uniform distribution of probes under the effect of the flow. This is achieved through a photobleaching reaction which locally destroys a fraction of the fluorescent molecules if they are irradiated whith a high intensity beam of light with a wavelength in the absorption band of the probe. The photobleaching reaction is performed before turning on the flow. Several bleaching patterns can be used. In the N.F.V. technique, the photobleaching pattern is obtained by interference fringes localised in the immediate vicinity of the bottom surface. These fringes are formed by the crossing of two laser beams of equal intensity and wavelength )~3 (obtained by a polarisation based interferometer described in more detail below). The angle ot between the two incoming beams defines the fringe spacing i" i-

~,0 The angle of incidence on the bottom surface, 0i, defines the vertical spatial 2 sin a/~2"

resolution of the experiment. If 0i is larger than 0c, total internal reflection occurs" 0c = sin-'(n~-//~ ). Then, anon-propagative evanescent wave exists in the liquid, with an \ / L I Ij

exponentially decaying profile along the z direction, and a characteristic decay length A, the penetration depth. The investigated area has an approximately elliptic shape, with a typical size of .5 x 3 mm (The useful part of the sample in which interferences are formed has in fact the shape of the intersection of the two laser beams with the bottom surface, i.e. the intersection of two ellipses (see figure l b)). To increase the sensitivity, we have used a spatial modulation of the position of the fringes as described by J. Davoust and L. L6ger [28]. To understand how this modulation is produced, we need first to describe the interferometer which produces the fringes. An Argon laser beam is split into two beams with crossed polarisations by passing it through a Wollaston prism. The angle between these two beams is fixed by the Wollaston prism. Their relative intensity depends on the angle between the polarisation of the incoming laser beam and the directions of the neutral lines of the prism. For an angle of 45 ~, the two outcoming beams have the same intensities. These two beams are rendered parallel to each other by passing them through a biprism, then the two polarisations are set parallel by letting one beam cross a half wave length plate. Finally the two beams are recombined by passing through a converging lens: as they are parallel, they recombine at the focus of the lens, which is adjusted to be exactly at the botom surface of the sample.In order to modulate the position of the fringes a phase shift is introduced on one of the two interfering beams via a Pockel cell placed in front of the Wollaston prism, with its neutral axis parallel to those of the Wollaston prism. The amplitude of the modulated phase shift is controlled by adjusting the modulated voltage applied to the Pockel cell, thus one can make the fringes oscillate around their equilibrium position with an amplitude of half a fringe spacing, at the frequency F. The experiment is performed in two steps: 1) the position of the fringes is held constant I no spatial modulation) and the full power of the laser is shined for the bleaching period (typically 50 ms), thus printing into the sample, over the thickness A, a periodic distribution of fluorescent probes, with the periodicity i 2) the power of the laser beam is attenuated by a factor 5x 103 , so that no appreciable bleaching of the fluorescent probes can occur during the whole measuring period, and the spatial modulation of the fringes is turned on. The

340

fluorescence intensity, collected with a photomultiplier, is then the superposition of a DC. level, an F and a 2F components[28]. The two modulated components are proportional to the product of the bleaching and the reading intensities both having a vertical distribution in e -z/A, we are thus probing a slab of thickness A/2 above the solid surface.In the conditions of the experiments presented here, A is typically 1000,~ and the vertical distance probed is comparable to the radius of high molecular weight polymer chains. The F and 2F signals exhibit various features depending on the flow pattern: i) without flow, the spatial distribution of fluorescent probes produced by the bleaching pulse relaxes towards equilibrium through diffusion, leading to an exponential decay of the 2F signal and to a zero F component; ii) with pure shear flow and no slippage, the original bleached fringe pattern is progressively tilted, with a tilt angle increasing linearly with time. This distortion of the bleached pattern leads to a monotonic decrease of the 2F signal, and to the onset of the F signal. At long times both components go to zero; iii) if slippage occurs, with a slip velocity Vs, both the 2F and the F signals oscillate at the frequency v = Vs/i, with the same amplitude, and in phase quadrature [28]. These oscillations in both the F and the 2F signals are a clear signature of slippage. Vs is obtained by either measuring the time period of the oscillations (Vs = i/T) or their frequency, v, through a Fourier transform analysis of the signal. The technique is limited at high velocity, when the characteristic time of the data acquisition system is comparable to l/v, leading to a decrease of the signal to noise ratio. It is also limited at low velocities, when the bleached pattern relaxes by diffusion faster than the appearance of the oscillations due to the flow. The easily available range is typically l O-2~trn/s "

' '" "I -

' '"'=I

' ' " "5

' ' ' "~

: o

-,

qo

"~

~o

V*

101

O0 O

"~

.

_ i

[ l

0~ n , innnni

10 -1

" , n nnlml~

n , tlln~

I l ..ml

101

' u,

.

1'/'" 9

"

10 0

1 0 -2

' '"=I--' ' '"'I

b

=

10 2

E

10 3

I,,,,,

~O

,.hid

'1

~

/

, ,inn|

, ,n,n "I

.

~

9

=,

10 1

10 3

, ,lind

10 .2

Vt (l.tm/s)

10 0

, in.,-]

, . u . "I

10 2

n inn,n,

10 4

V s (pm/s)

Figure 8. Results obtained for a polymer melt of molecular weight 9.6 105 flowing on a silica surface covered with a pseudo brush made from a melt of molecular weight 1.93 105. a) Vs as a function of Vt; b) slip length b as a function of Vs. The friction in the low shear regime is comparable to that obtained on a low density surface layer, but the critical velocity V* is much larger.

3. MOLECULAR MODELS AND DISCUSSION: 3.1 The ideal surface- de Gennes's conjecture: Fifteen years ago, de Gennes suggested [18] that high molecular weight polymers, when in contact with a smooth non adsorbing wall, should always display a slip boundary condition, with an extrapolation length b depending only on the molecular weight of the polymer chains, and able to become very large, i.e. to render the slip easily visible macroscopically. This relies on the following remarks: monomers are in contact with the wall; the constraint at the wall, a, is related to the friction, k, between these monomers and the wall, by k = o/Vs; k is characteristic of monomers/wall friction, and is expected to be the same for a liquid of small molecules identical to the monomers or for the polymer liquid. The stress G can also be evaluated in the liquid: cr = 1] dV/dz z=O = q Vs/b if slip occurs with the extrapolation length b. Then, comparing the situation of the liquid of monomers, for which b = a, and the polymer, one expects b = a q/rio, for the polymer case, with 1"1the bulk viscosity of the polymer melt, and qo the viscosity of the equivalent liquid of monomers. In the reptation approach, the viscosity is q = q0 p3/Ne2, with P the polymerisation index of the polymer chains, and Ne the average number of monomers between entanglements [40]. For high molecular weights polymers, with P = 104, Ne = 10 2 . and the size of a monomer a = 3A, one gets b = 3 cm, i.e. huge slippage. Thus. high molecular weights polymers should slip whatever the shear rate. in a linear slip regime, i.e. with an extrapolation length b independent of the flow regime. The extrapolation length, b, and the magnitude of the slip velocity at a fixed shear rate, should strongly depend on the molecular characteristics of the polymer. These features do not seem to be observed experimentally. It has been progressively recognised that practical experimental situations could be very different from the ideal surface postulated in the simple de

349

Gennes'analysis, and that many additional effects could complicate the description. In many systems where the polymer melt is not far from its glass transition temperature, attractive interactions between the solid surface and the polymer can lead to the formation of a thin glassy layer close to the surface. It was first thought that such a glassy layer could block slippage at low shear rates [41 ], due to entanglements between chains pertaining to the glassy layer and chains in the liquid melt. The role of entanglements between chains immobilised at the surface and chains in the melt on wall friction has thus been investigated, starting from simple systems where the immobile chains are independent of each other to go towards more realistic and more complicated situations where the surface chains are interacting with each other and with the melt [23 - 25]. Obviously, the roughness of the solid surface can also play an important role, modifying the flow pattern and possibly deforming the polymer chains, thus reacting on the flow characteristics. This is certainly an important issue, from a practical point of view, but it has not yet been investigated in detail theoretically.We shall concentrate here on the effect of polymer molecules attached to an otherwise smooth and fiat wall, and analyse successively the two situations of independent adsorbed chains and of dense immobilised layers. 3.2. Role of polymer chains anchored to the wall at low surface densities A few polymer chains attached to the surface have a drastic effect on the boundary condition for the flow velocity: they strongly reduce the magnitude of slip at low shear rates [23], as they increase the surface friction compare to an ideal surface. However, polymer chains are easily deformable objects, and under the action of the shear forces, the surface chains tend to elongate and to disentangle from the melt at high enough shear rates. This aptitude to deform in the flow is responsible for the appearance of a shear rate threshold above which strong slip, comparable to what is expected on ideal surfaces, is recovered. The magnitude of this shear rate threshold has been related to both the molecular weights of the surface attached chains and of the flowing polymer, and to the surface density of anchored chains by Brochard and de Gennes [23], modelling the friction between the surface chains and the flowing melt, and the deformability of the surface chains. One surface chain (polymerisation index N), anchored on the solid by one extremity, experiences a friction force due to all the other flowing chains from the melt (polymerisation index P) which are entangled with it. The N chain elongates under the action of this friction force. Two delicate questions have to be answered in order to estimate the friction force exerted by the melt on one surface chain: first, one needs to evaluate the relative velocity of one melt

v (-xD

Figure 9. Schematic representation of the molecular process allowing the flow of the P chains when entangled with the surface N chains.

350

chain and of the surface chain when the local velocity of the melt at the surface is Vs, and secondly, one has to estimate the number of independent free chains from the melt which indeed are entangled with the surface chain. In the reptation picture, the process is schematically represented in Fig. 9: a P chain is entangled with the N chain fixed at the wall; in order to allow for their relative motion with velocity Vs, one extremity of the P chain has to pass close to the N chain and relax the topological constraint, in a time shorter than the time 1: it takes for the N chain to travel parallel to the surface over a distance comparable to the tube diameter, i.e.: 1: = (aNel/2)/Vs . The P chain has thus to move inside its own tube at a velocity Vtube, higher than Vs"

PaN , Vtube - Ltub-------~e- Ne

P

Vs The amplification factor P/Ne is responsible for a strong dissipation and this is the origin of the suppression of slip at low shear rates when chains anchored to the solid wall are present. Knowing Vtube , the friction force associated with one P chain can be estimated" the dissipation in the relative motion of the P chain along the N chain is TS = P~Vtube 2 = fvVtube, with ~ the local monomer-monomer friction and fv the friction force between the two chains. One has thus fv = arlpVs, with rip the bulk viscosity of the melt of P chains. The total friction force experienced by the N chain depends on the number of P chains which are entangled with it. This is one of the delicate problems of the reptation picture: the total number of entanglements between the N chain and its neighbours is N/Ne, and is fixed as soon as the average number of monomers between entanglements Ne is known, but it is not obvious whether or not some P chains have more than one entanglement with the N chain, with the result that the number of P chains trapped by one N chain may be different from the number of entanglements. This point has been strongly debated in the recent years (see the corresponding paragraph in the chapter on the reptation model in this book), and recently reanalysed and applied to the slippage problem by C. Gay et al [24,251: two different regimes in the way the number of trapped P chains, X, depends on N have to be distinguished" i) if N < Ne 2 , X = N//N , and each P chain has on the average one entanglement with the N chain; ii) if e

N > Ne 2, X = N 1/2 and one P chain has more than one entanglement with the N chain. With a typical Ne value of 100, only very long grafted chains are in the regime with more than one entanglement per trapped P chain. In the simple regime with N < Ne 2, the total friction force experienced by the N chain is N Fv = r l p a ~ e Vs. Following the Pincus picture [42], an elastic restoring force develops, kT Fe] - - ~ , D with D the diameter of the associated Pincus blob. Equating the friction and the elastic forces, one gets kT N = arle Vs 9

D

7 e"

above the velocity V 1 for which Fv = kT/RN, D decrease with Vs and the N chain elongates along the flow. When the velocity is increased, the N chain elongates more and more and the

351

diameter of the Pincus blob progressively decreases, down to a value comparable to the average distance between entanglements in the melt. One thus enters into what has been called the marginal regime: if the velocity is further increased, the N chain tends to disentangle. Then the friction decreases and can no longer balance the elastic force associated with the large elongation" the N chain thus tends to recoil. In fact, above the critical velocity V* for which D becomes comparable to the average distance between entanglements, the elongation remains locked at the value it had at V*, for a large domain of surface velocities. These ideas can be used to model the flow behaviour in the presence of surface anchored chains at low surface densities ( i.e. for N chains in the "mushroom" regime of grafting characterised by a number of surface anchored chains per unit area,v, related to the dimensionless surface density of chains Z by Z = va 2, such that VRN2 0, or again q > q*. It is t h e n possible to d e t e r m i n e t h e corresponding values of: - APt (cf. Fig. 3a) by means of relation (4b) ; - APe (cf. Fig. 3b) by means of relation (5b) ; - APc (cf. Fig. 3c) by m e a n s of relation (7).

369 By introducing the pressure loss APc obtained in this way into relation (3), it is possible to calculate the stress I;R associated with the flow rate q considered. Relation (2) can then be used to determine the value of UR corresponding to this same flow rate, thus indicating variations in TR as a function of UR, and hence the friction relation for the fluid under the flow conditions considered. This method was applied to two PDMSs (LG2 and LG3), to the PB and the LLDPE (cf. Table 1). The values of"Kec", "Ked", "Ktc", "Ktd", "Kc", "nec", "ned", "ntc", "ntd", "n", and the value of q*, determined from the experimental curves obtained for these fluids are given in Table 3. This table also contains the value of the shear stress at the wall for flow rate q*, denoted x*. By choosing these fluids, it was possible to obtain a comprehensive view of the friction curves for polymer melts with slip, over a significant r a n g e of slip rate at the walls. Figs. 4a to 4d demonstrate: - The existence of a static friction stress, corresponding in fact to the flow rate q* on flow curves. Thus in the conditions used in this study, the polymer melts only slip above a critical stress threshold. They adhere when at rest. 1,o (10

R

5

Pa)

0,8

A

0,6

A

j

0,4

---"i,.5-J

-'-

Gum_LG2

9

A L/D = 10/0.5 Q L / D = 20/0.5 0,2

9 L / D = 20/2 R e l a t i o n (10)

0,0, 10 4

9

'

'

'

'

1 ~ 1 -

10

'

-3

"

'

' ' " 1

1 0 -2

V R (m/s) '

'

'

'

' ' ' 1

10

,

"1

,

1-

,

,,

,,

10 ~

Figure 4a " Friction curve of PDMS LG2. x* is the static friction stress (Table 3). - At low slip velocities, the stress at the wall decreases when the slip velocity increases, which is a current situation in tribology. This decrease corresponds to the flow regimes preceeding the m i n i m u m obtained on the flow curve. Consequently, it is not possible to deduce shear stress value in the decreasing p a r t of the friction curve from the flow curves, since they correspond to the oscillatory flow regimes. Following this decrease, the shear stress at the wall increases again with slip velocity for flow regimes corresponding to portion

370 (DF) on the flow curves. Note that in the case where the flow curves exhibit two instability zones, the corresponding friction curve exhibits two minima, the last one being followed by a rising branch with a high slip velocity corresponding to portion (HG)on the flow curves. The friction curve depends slightly on the degree of e n t a n g l e m e n t in a given family of polymers (comparison of Figs. 4a and 4b). In addition, in the case of the LLDPE, the friction curve does not appear to depend on t e m p e r a t u r e (Fig. 4d).

9

- -

_

.

(105 Pa) R 0,8

~, (5/0.5)

LG3"

Gum

o I A ) = 5/0.5 9 I./D = 20/2

0,6 x, (20/2)

00 0,4

ll E l l

mo

ooOOO

9

o

0,2

U R (m/s) 0,6 10 -4

9

'

,

,

,'

,

9

10 -3

"v

,"

,

9

9

,

i

'1

10 -2

10 -1

Figure 4b 9Friction curve of PDMS LG3. ~* is the static friction stress (Table 3). Thus it appears that, generally speaking, the run of the friction curves obtained for the various polymer melts is characterized by significant non-linearity. On a semilog scale, these curves have in particular a bell shape, which is reminiscent of the work carried out by Moore in 1972 [31] on friction using slightly r e t i c u l a t e d elastomers. These also show t h a t adherence and h y s t e r e s i s are the m a i n mechanisms governing slip phenomena. This may be explained by the fact t h a t the polymer melts used in this study are highly entangled. In addition, these curves have two minima corresponding to the two oscillation regimes in the instantaneous pressure. It should be noted that, in the case of the PDMSs, the small quantities of fluid available for the s t u d y were insufficient to r e a c h flow r a t e v a l u e s corresponding to the second minimum, if it exists. It is also possible to characterize the polymer-wall slip by representing variations in the extrapolation length b as a function of the slip velocity. It should be recalled t h a t b is defined as the ratio of the slip rate to the shear rate at the wall [9]. With the assumptions made in this study (power law, etc.) relation (2) can be used to determine the shear rate at the wall for flow regimes including slip, i.e.:

371

3,0

5 R

Pa)

(10

O

2,5

O

O $

9 8o

O

2,0

@0

0 o~

O

9

1,5

1,0" L / D = 5/0.5

o

9 ][.JD = 20/2

0,5

U

R

...... .

0,0

'

'

(m/s) 9

'

'

"

9

'

1 0 -3

I

'

9

9

.=,

.

'

1 0 "1

1 0 .2

F i g u r e 4c" F r i c t i o n c u r v e of PB.

5 Pa)

I; R ( 1 0

05

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